The French "printemps" which I suppose is where "printempo" comes from is itself derived from Latin "tempus primum" (the first period, i.e. season).

Esperanto kinda has both those roots (tempo=time; principa=fundamental/basic), so I suppose in a way, yes, but I think Zamenhof probably just took his word from the French as it is.

vintr/o and printemp/o are separate words (unrelated to e.g. vi/n, tro, or pri, temp/o, or po),
in a similar way as there, supposed and other are separate words unrelated to the, re, pose, and her.

(but yeah, it can at times be a bit confusing (not "con"+"fusing", though ;-) ))