As many here have already said, if this Vegito is the one from the Buu arc, then Omega shouldn't have much trouble, but if it were a Potara fusion between Goku and Vegeta during the GT arc, then a SSJ4 Vegito would annihilate Omega Shenron ten times over with more ease than it took SSJ4 Gogeta to curbstomp OS for the ten minutes he existed. That said, Vegito, be it SSJ1 or SSJ4, wouldn't have the problem of a time limit, even though he'll be likely to act just as cocky as SSJ4 Gogeta (if not more).
Hell, if the latter is allowed in this thread, then Vegito, with the combined might of his jawbreaker form, would win in a multi-fold overkill.