The question is somewhat moot in that the encounter of Muhammad with this Christian monk Sergius/Bahira is almost certainly apocryphal. As with many such details purporting to be from the life of the prophet our sources for this encounter come from centuries after the prophet's death, namely, the Sira of Ibn Ishaq, Tabari and Ibn Sa'd. The encounter serves a clear polemical purpose in establishing that Muhammad was foretold in a tradition that is otherwise absent of such a foretelling (in contrast with Jesus, who could be said to be the Messiah foretold in the Jewish scriptures). Muhammad's meeting with the monk aged 12 is the same age that Jesus sat with the doctors in the temple. Moreover the lack of detail in the traditions, extending down to the monk/s name, who the adult in the story was (Abu Bakr or Abu Talib), what it was that made the monk notice Muhammad's special providence, etc, cast further doubt on the veracity or credibility of the account.

That being said, as far as I'm aware the early Islamic tradition doesn't specify a sect for the monk. Later texts seem to suggest that he was viewed as a kind of sect-less "pure" Christian. Christian polemics on the other hand describe him as either a Nestorian, or an iconoclast, or as a Jacobite, or as an Arian, or don't specify.

though it doesn't seem to be specified where

The meeting is supposed to have taken place at Bosra, Syria where there is a "monastery of Bahira."

do we know what kind of Christianity that was popular around Arabia during Muhammad's time?

Here there was a division between Eastern and Western Arabia, given the division of the sphere of influence between the Byzantines and Sassanians. In Eastern Arabia, Nestorian Christianity would have been predominant (as with the Lakhmid client tribe of the Sassanians). In the West, despite being converted by the Byzantines the predominant form of Arab Christianity would have been monophysitism rather than the imperial Chalcedonian church.

There were Miaphysite Arabs helping Rashidun armies during Battle of the Bridge (against Sassanid).

The Arab armies included a significant number of Christians. I don't think it would be useful to talk about Christian Arabs "helping" a Muslim army-- they were the same army. Conceptually the distinctiveness of a separate "Muslim" identity as opposed to "Muslims" being simply pre-eminent among the wider community of monotheist believers did not exist, hence the title "amir al-muaminin", Commander of the Believers.

Source-wise, any accounts or descriptions of battles in this time period should be treated with extreme skepticism. Any source that purports to "elaborate further" should probably be treated as less reliable, not more, given the time between the composition of this history and when the purported events occurred.

Source wise

For Bahira, see the entry in The Encyclopaedia of Islam For the religion of pre-Islamic Arabia see Robert Hoyland's Arabia and the Arabs: From the Bronze Age to the Coming of Islam. For the conquests see either (or both) Hoyland's In God's Path or Fred Donner's Muhammad and the Believers.

edit: forgot to finish a sentence.