Strictly speaking, I think Roman sun-worship was henotheistic (i.e., believing in the existence of multiple gods, but worshiping one of them as supreme) rather than monotheistic.
As for Christianity, Constantine's conversion was a major boost to it, although TBH I think the importance of imperial patronage has often been exaggerated. For one thing, Christianity had done quite well for itself over the previous three centuries; for another, most of the fourth-century emperors were Arians, but Arianism itself was a spent force in the Empire by the end of century, indicating that imperial support wasn't the be-all-and-end-all of religious success. So, if Constantine hadn't converted, I'd expect Christianity to continue its previous trajectory of steady but modest expansion; it won't do as well as it did IOTL, but it will still be a major religious force at the fall of the Empire.
Speaking of which, it's interesting to consider what a different religious makeup would mean for the fall of Rome and the following centuries. Of course, without a Christian Roman Empire, the religion would have less cultural cachet, so the barbarians might be slower to convert; on the other hand, assuming that secular education declines much as IOTL the Church would be left as the only real source of literate administrators, so we might see the post-Roman rulers patronising the Church and even converting themselves. In the long run, then, Western Europe might end up Christianising anyway, although the different circumstances would have a potentially major impact on the way people view their past -- in particular, later writers would probably remember Rome as this great, oppressive tyranny overthrown by pro-Christian liberators, rather than as the ideal empire against which other states should measure themselves.
As for Constantine himself, without his religious reforms he'd probably be much less well-known than IOTL. Most likely he'd be remembered as just another late antique emperor. Maybe he'd get credit for completing Diocletian's transformation of the empire -- it's often difficult to tell whether a given reform was enacted by Diocletian or Constantine, although Constantine's importance in religious history has tended to overshadow his other contributions IOTL. Outside of history geek circles, though, I don't think he'd be very widely known.