Armenia was the first country that adopted Christianity as their primary religion in ~300 A.D. But what kind of Christianity do (did) they practice, as it is not of Orthodox or Catholic Christian variations?

This is a bit tricky to answer, but I'll give it a try. An important thing to note is that "Orthodox" or "Catholic" are not useful term for describing the late-antique Church; instead, I think it is better to see the Church as composed of various factions who agreed on some doctrinal matters but disagreed on other issues. Nor should we see Armenia as entirely Christian or of a certain sect, as the conversion of some of the elites did not mean that the entire kingdom embraced the religion. So when I talk about the various doctrinal positions of the Armenian Church here, do keep in mind that the actual beliefs of the Armenians were varied and that some of the common people no doubt held very different beliefs to their leaders - we just don't know very much about them, since literary sources were largely written by the elite.

Armenia, as you noted, made Christianity its official religion in the fourth century and it seemingly adopted a form of Christianity that agreed with the formulation chosen at the First Council of Nicaea in 325. The Armenian clergy was seemingly Nicene in character for the next century and took part in the next two ecumenical councils too (Constantinople I, 381 and Ephesus, 431). Note however that the Church within the empire was deeply divided between followers of different doctrines, with the homoian brand of Arianism and "Nestorianism" (which was really just a polemical term coined by partisans of Cyril of Alexandria suspicious of the Syrian Church's theological position) being two major bones of contention. Armenia wasn't very involved in these disputes, but it was geographically and culturally closer to the Syrian world, so some members of the Armenian Church must have been aware of and partook in the wider doctrinal disputes of the fifth century. A distinctive doctrinal position would also enhance the Armenians' political position, since it would create a sense of difference and independence from their more powerful imperial neighbour. These factors I think made the Armenians' later break with the Roman Church more understandable.

This break came after the Council of Chalcedon in 451, in which the Christological position that Christ possessed two natures bound together in one divine hypostasis won the debate. Though the Armenians did not officially endorse the miaphysite position (essentially that Christ had one nature from the union of the human and the divine) until the early sixth century, I think it would be reasonable to say that the Armenians must have been moving towards that position already. After all, many people in Palestine and Syria were outraged by Chalcedon and the emperor at Constantinople had repeatedly tried to force through a compromise formulation (and had failed every time). Though this is unstated in the sources, the Armenians must have been involved in this process as well; from their neighbours the Georgians we for instance hear of Peter the Iberian, a miaphysite prince-ascetic active in Palestine.

The Armenian Church seemingly held on to the miaphysite position for the next century, though the Romans and the Persians did attempt to force the Armenians to embrace Chalcedonian Christianity and Zoroastrianism respectively; these persecutions were of course wildly exaggerated by the sources, so the miaphysite Church was probably never threatened too much. Nor could the Armenians fully condemn Chalcedonianism, since to do so would anger the emperor at Constantinople. Indeed, the Roman emperor Justin II even intervened to help Armenian rebels fighting Persia in 572. Evidently, the divide between Chalcedonians and miaphysites was not yet so wide that it threatened their mutual sense of Christian unity. There was an explicit condemnation of Chalcedonian Christianity in 607, but this only occurred after the conquest of Armenia by the Persians (as they were aware of internal Roman divisions and were quite happy to exploit it in order to make themselves more popular amongst their new subjects). This was of course reversed when the Romans defeated the Persians in 628.

The rest of the seventh century was a period of doctrinal flip-flopping within the Armenian Church, as the Roman emperors imposed monotheletism (another Christological formulation) on the Armenians at least twice, which was promptly abandoned for miaphysitism whenever Roman influence was weak - which was quite often due to the rise of Islam. Still, monotheletism was popular amongst some Armenians, as a later emperor with Armenian roots imposed it on the empire when he usurped the throne. I know very little about the Armenian Church after the seventh century, but miaphysitism is still the defining characteristic of the Church, one that differentiated it from the modern Catholic/Greek Orthodox Churches. This position is also held by other churches like the Coptic Church of Egypt, which together with the Armenians are known as the Oriental Orthodox Churches. Obviously other beliefs did exist, with the famous Paulician heresy being the best example, and the Romans had continued to try to reconcile the Armenians with Chalcedonian Christianity for centuries afterwards, but the Church we now know as the Armenian Apostolic Church was largely forged during the controversy over Chalcedon.