The claim to analyze and compare here is that for 'mentioning Jesus' we have many more existing manuscripts that are much older than many and most manuscripts for almost any other historical person of antiquity. And that overly specific claim is correct.
If we just compare Caesar with Jesus, in that way, Caesar almost loses. "Almost", since the bulk of manuscripts by Caesar or about him undoubtedly are lost now. However, one tiny and exceedingly rare counterexample is shown below.
The oldest fragment for 'Jesus' is the recto of the Rylands Library Papyrus P52, 𝔅52 / 𝕻52:
ΟΙ ΙΟΥΔΑΙΟΙ ΗΜΕΙΝ ΟΥΚ ΕΞΕΣΤΙΝ ΑΠΟΚΤΕΙΝΑΙ
ΟΥΔΕΝΑ ΙΝΑ Ο ΛΟΓΟΣ ΤΟΥ ΙΗΣΟΥ ΠΛΗΡΩΘΗ ΟΝ ΕΙ-
And this is dated to be an 'original copy' from between first and 3 century. Note that the actual word 'Jesus' is missing in this piece, but assumed to be 'has to be there' from later and more complete papyrii. For a direct evidence of that word we need to look at the slightly younger 𝕻90, in which we find the part of the Gospel of John that mentions Jesus wearing a crown thorns.
The list of extant biblical manuscripts is long.
The New Testament has been preserved in more manuscripts than any other ancient work of literature, with over 5,800 complete or fragmented Greek manuscripts catalogued, 10,000 Latin manuscripts and 9,300 manuscripts in various other ancient languages…
Compared to for example Caesar's Commentarii de bello Gallico one might argue that this is seen as at least being from Caesar himself as the original author.
But unfortunately the oldest known manuscripts for this work date only to the ninth century.
Caesar's literary fame in modern times is founded solely on the comm., which were published posthumously in the Middle Ages as part of the Corpus Caesarianum; rarely read in antiquity, they survived in a late-antique MSS (corrected in the BG ?), which in the Middle Ages was transmitted in a few copies, mainly in France.
— Will, Wolfgang (Bonn) and Rüpke, Jörg (Erfurt), “Caesar”, in: Brill’s New Pauly, Antiquity volumes edited by: Hubert Cancik and , Helmuth Schneider, English Edition by: Christine F. Salazar, Classical Tradition volumes edited by: Manfred Landfester, English Edition by: Francis G. Gentry. First published online: 2006 doi
And that caesarian literary heritage of bello Gallico is summarised as:
There are 6 early witnesses to the alpha family. Two derive from a common lost ancestor: these are:
Amsterdam 73, 2nd quarter of the 9th century, written at Fleury (=A)
Paris lat. 5056, 11-12th century, written at Moissac (=Q)
The remaining four derive from another now lost ms:
Paris lat. 5763, 1st quarter of the 9th century, French, later at Fleury (=B)
Vatican lat. 3864, 3rd quarter of the 9th century, written at Corbie (=M)
Florence, Laur. Ashb. 33, 10th century, possibly French (=S)
British Library Additional 10084, 11-12th century, probably from Gembloux (=L)
Some 75 mss later than the 9th century have been listed by Virginia Brown, who has classified them into groupings tentatively.
The Klotz edition of 1950 used 8 mss, although at least 3 of these are now considered to be non-primary. The five are:
- Florence, Laur. 68.8, basically 10-11th century, probably Italian, once the property of Niccolo Niccoli (=W)
- Vatican latinus 3324, 11-12th century, possibly French (=U)
- Paris lat. 5764, 3rd quarter of the 11th century, French (=T)
- Vienna 95, 1st quarter of the 12th century, probably from Trier (=V)
With Paris 5763:
— ark:/12148/btv1b8426038x Bibliothèque nationale de France. Département des Manuscrits. Latin 5763
The same applies to other authors writing about Caesar: their works are sometimes known to us today but, the historical literary transmission of ancient texts is much sparser and temporally more removed from the actual time they lived than New Testament writings. Most of the manuscripts containing information about 'profane' historical figures are high and late medieval copies.
one of the best contemporary primary sources for providing historical proof of Caesar's existence would be to point to more robust than papyrus material remains, like inscriptions in stone or metal, perhaps even a coin that the man had issued himself, sometimes with his name or even head on it.
A thing we're told Jesus didn't care for that much.
This answer so far avoids 'the question in title on purpose'. The 'best primary sources for the existence of Caesar' is really a XY-problem. Answering the title question as is, would mean that 'claimant to be disproven' and OP will talk past each other, largely.
We have comparatively 'hard' objects which point to Julius Caesar directly, like CIG 2957:
Honorary inscription for Gaius Iulius Caesar by poleis, [demoi], and ethne (of Hellenes) in Asia; 48 BC; found at Ephesos: CIG 2957; LW 142; Syll3 760; Tuchelt, Frühe Denkm. 141; *IEph 251.
However, there probably is one striking example fulfilling all the requirements needed to get a surviving manuscript from the time of Caesar's assassination:
Fata mihi, Caesar, tum erunt mea dulcia, quom tu / maxima Romanae pars eris historiae / postque tuum reditum multorum templa deorum / fixa legam spolieis deivitiora tueis.
This is Latin poetry written by Gaius Cornelius Gallus (c. 70–26 BC). Coincidence: this is classified as PQasrIbrîm inv. 78-3-11/ (L1/2) and also portrayed as
The Oldest Surviving Manuscript of Latin Poetry Was Discovered in Qasr Ibrim — Circa 50 BCE to 25 CE.
"The book can be dated from its archaeological context, more precisely (c.50-20 B.C.) or less precisely (c.50 B.C.- A.D. 25). It therefore provides one of the few fixed points in the early history of Latin literary scripts."
But what the claim brought forward alludes to is the undeniable fact that for a text from classical antiquity the literary transmission of New Testament writings is indeed remarkably well, in many cases going back far with little variants and alterations as evidenced from the oldest copies we did find. The vast body of classical literature seems to be present in our finds only in often much younger copies, and with fewer copies. Some of those then also with heavy disagreements between them.
For literature from classical antiquity the claim has a sizable kernel of truth to it. Many of the otherwise seemingly well known historical figures can be seen as being harder to ascertain as being 'real' than the main protagonist of the gospels, if going by oldest surviving manuscripts alone. Unfortunately, that on its own doesn't even prove the existence of said protagonist without remaining doubts…
At the beginning of this answer, we read "almost any other historical person of antiquity". Since that claim is often phrased as "better evidence for (absolutely) any other historical figure", and meant to convey 'any source material at all included' we see that this special viewing angle is then taken to extreme exaggerations that is simply untrue.
A more fitting comparison would be to compare literary transmission for Sokrates and Jesus and the value of this transmission history for 'judging things'. Known basically only through writings about him, mainly by his disciple, whose writings are of quite late provenience, etc. That would not prove anything either on its own, but illustrates the sometimes differing scales that are applied in these matters in disputes.