Однажды он сказал себе: "Сегодня я напишу ей письмо о Кате. Или просто позвоню ей." I'd say that perfective is more appropriate here, as basically you ar etrying to say that you will have it done by the end of today. Буду писать means that you will be in a process of writing but it does not suggest that you will have it completed by the end of today. I hope it makes sense. Cheers

This variant:" Сегодня я напишу ей письмо о Кате. Или просто позвоню ей" is correct. And it sounds naturally.You should use here perfective aspect because you will write this letter only today and only one time. And you want to finish it today.If you want to have some result you should use perfective. The second variant is theoretically possible and it is correct if you want to emphasize that it will take a lot of time and you don't know whether you will finish this work.