There have been several travels in boats reconstructed from the relevant times. One that I can find good records of is the boat known is Aifur. It travelled in 1994 from Sigtuna in Sweden to Novgorod. This took 41 days.
In 1994, the Aifur crossed the Baltic Sea and sailed up
the rivers Neva and Volkhov to Novgorod. Distance covered was 1382 km.
The effective time was 307 hours, of which sailing time 192 1/2 hours
and rowing time, incl. manual towing, 114 1/2 hours.
In 1996 they continued from Novgorod. This was during a year with very little water in the rivers, so this can be counted as a sort of "worst case":
covered was 1568 km. The effective time was 415 1/2 hours, of which
sailing time 113 1/2 hours, rowing, incl. manual towing, 264 hours and
manual towing over land, 38 hours.
The total time was 113 days. It might have been faster other years, with more water. From Odessa it should be a fairly quick sailing trip over the Black Sea to Istanbul, but the only period-authentic expedition I can find that ended up in Istanbul went via Poland (because of Politics).
More reading: http://web.comhem.se/aifur/vikingvoyages.pdf