A usable bicycle depends on lightweight and low friction power transmission. Probably you would therefore need to first invent ball bearings ( and to do this you need some metallurgy).
So I believe that a time traveler could probably bring enough technology stack with him, that he could build a working demonstration bike in ancient Rome. I am a lot less sure, that there is a consistent alternate history in which Cesar is riding a bicycle.
Penny Farthings and velocipedes operated with a direct drive, where pedals were attached directly to wheels. This was the reason for the large wheels of the penny farthing (without a geared chain-driven transmission there was no other way to step the drive ratio), and created the somewhat obvious limitations on braking and inability to freewheel also present on modern-day "fixies".
