I would say - I was so caught up in the video *that* i did not notice the phone ringing.

It depends on what you're trying to say. If you mean you didn't hear the phone WHILE it was ringing (past progressive tense) you could say, "I was so caught up in the video that (instead of "so") I did not notice the phone ringing." If you want to say after the phone was done ringing that you realized you had missed the call, you could say, "I was so caught up in the video that I didn't notice that the phone had rung" (past perfect tense). The first sentence indicates that you missed the call during the phone's ringing, whereas the second sentence indicates that you realized after it was done ringing that you had missed the call.

You could say any of the following: "i didn't realize the phone was ringing" or "i didn't realize the phone had rung" or "I didn't realize the phone rang"