How can I analyze (not summarize) the following from Life of Constantine by Eusebius of Caesarea? "For as soon as he was made acquainted with the...

The key strategy you need to follow in analyzing a passage like this is thinking about its context. To discover the antecedents of pronouns, you need to look to the immediately preceding materials to discover their referents. This is not a difficult process; simply pasting the quotation into a Google search box will pull up the chapter from which this was excerpted, namely Chapter V.—Of the Disagreement respecting the Celebration of Easter.

Thus this is not really an anti-Arian statement but rather one on the subject of the Paschal or Eastern controversy, concerning how the date of Easter is to be determined. Syrian Christians were the main opponents to Constantine and the Council of Nicaea on this issue. 

The main ambiguity here is that there appears no obvious "secret adversary" mentioned. Chapter IV discusses the Arians and then Chapter V moves on to the Paschal Controversy, which actually started much earlier, really gaining traction in the late second century. No one individual is mentioned in regards to the Paschal Controversy though; all we know is that the Council issued statements which attempted to create uniformity in the date of Easter across Christendom and failed (Orthodox and western Christians still calculate it differently). Eusebius thinks this discord a great evil, describing it as a "most virulent disorder" and saying:

... the people being thus in every place divided in respect of this, and the sacred observances of religion confounded ... no one appeared who was capable of devising a remedy for the evil, ... Constantine appeared to be the only one on earth capable of being [God's] minister for this good end.

The main evil mentioned in the passage is discord and disunity, and thus you could make a case for that being the enemy Constantine is opposing. "War" here is metaphorical; Roman emperors did not begin sending armies against Christian heretics until the early fifth century.

For your analysis, what you might want to focus on is the way that Eusebius appears to consider church unity and authority so important, at times nearly seeing uniformity and strong central authority as a goal in and of itself.