Did anyone write about Jesus at the time at which he was alive?
I can't find any sources from the time in which he was alive after a quick preliminary search.
No, nobody did. Despite the fact that literacy was relatively high amongst the Jews at that time. And we have several historians living there or in the area in those days. Jesus was mentioned outside the gospels a few times, but those lines are generally seen as either pious forgeries or out of context.
Josephus mentions Jesus, but it doesn't fit in the text and contradicts Josephus own religious ideas.
Pliny the Younger mentions Christians, as followers of a certain Crestos or Christos. He doesn't go into the existence of Jesus, merely asking the emperor what to do with his followers. Pliny lived a century later, and by that time there were Christians. It doesn't prove Jesus lived, it proved Christians existed.
Now, since we're in the realm of religion here, I always wondered why nobody outside the bible mentions that the dead walked the streets of Jerusalem upon the death of Jesus. (Matt 27:50-54) You'd think this was something rather remarkable, worthy enough to write down for posterity or inform friends about. But nobody did. At least the governor should have informed his emperor about this rather unusual event. But even he didn't.
If the question is did anybody write about Jesus before his crucifixion then the answer is No.
If the question is are there Contemporary sources who wrote about Jesus then the answer is Yes.
While no sources are known who wrote of Jesus before the crucifixion there are writings by a contemporary of Jesus (4,6BCE–30CE) who talked with and wrote of people who knew Jesus. Also Josephus is not dismissed by most scholars as an independent verification of Jesus’s life.
Contemporary means living at the same time. The oldest books of the New Testament Romans, Corinthians, and Galatians take the form of letters from Saint Paul (4BCE–62CE) to early church leaders (dated 55, 56, 57 CE). And stories about St Paul’s experiences. St Paul was a contemporary of Jesus being about the same age as Jesus. Although Paul states he had never met Jesus before the crucifixion. He discusses meeting in his letters with St. Peter who was one of the apostles who knew and walked with Jesus.
The earliest letters of St. Paul predate Josephus by about 40 -50 years. But Josephus also contain two references to Jesus (books 18, 20) only one of which is contested. The contested verse is not thought to be entirely false insertion but to have been modified. The other verse from book 20 is not questioned for its authenticity.
Josephus on Jesus
Modern scholarship has largely acknowledged the authenticity of the reference in Book 20, Chapter 9, 1 of the Antiquities to "the brother of Jesus, who was called Christ, whose name was James" and considers it as having the highest level of authenticity among the references of Josephus to Christianity.
Mainly the Deuterocanonical books of the Old Testament:
(Prior to Paul most of the old testament had been translated into Greek (mid-3rd BCE), a language consumable to non Jewish Roman Citizens.).
None that we know of. And actually Jesus wasn't unique in that regard. There were a lot of ancient figures who we know existed, but nearly everything we have written about them was from after their deaths (Alexander the Great, for example).
I'm going to guess, though, that what you're really asking is whether or not Jesus existed. The short answer is yes. He did! Very few historians worth their salt would question that, Christian or otherwise. Evidence that someone named Jesus lived during the First Century in Judea, was crucified, and was the central figure of Christianity is so rampant as to be virtually indisputable. This evidence includes the written account of Tacitus, who mentions Jesus and who was so hostile towards Christians that his passage is almost certainly not a Christian alteration of the original text. Furthermore, the existence of Jesus is supported by models such as the Criterion of Multiple Attestation and the Criterion of Embarrassment.