Early on Campbell slept all the time. She was especially sleepy b/c she was anemic. Sometimes it felt as if she would sleep, wake up just in time to eat, then go right back to sleep. I think as long as he's eating enough, for right now it's fine. Of course, as always, you may want to check with your pedi, but ours said it was normal.
Trust me, in a few months, you'll miss this time....
My son was born at 33 weeks and came home after a month he is now 2months 2weeks old and also anemic. He eats and sleeps most of the day when he is at home and stays up maybe 30 mins during the entire day unless something is wrong. If we go out (someone elses house) he stays awake for most of the day.
My DD slept all the time. I used to have to wake her up to eat and there were times she never really opened her eyes while she ate so it was like one continous sleep with a bottle in between! It's only been a couple of weeks now that she'll stay awake for a few hours during the day.
DD slept a lot her first week or so home but she would start fussing and crying when it was time to eat.
I've noticed in the past week that she's awake more. Not a whole lot more, but for an extra 5 or 10 minutes after eating, or while we're holding her on the couch watching a movie or whatnot. Her due date was 12/31 so I think she's a bit more awake as she's nearing that day.