He is still tiny and you cannot really expect him to be on good routine yet.
However, when you say that at first wake ups he eats very little, I would try not to feed him at all these times at nights, if he is not truly hungry, it just makes a habit later on.
Does he gets DF?
Ideally it would be DF about 10 or 10.30 and then I would say feed him at about 3, when he wakes then.
If you dont DF I would suggest, feeding him at his first wakeup and then about 3 or 4 hours later and then only in the morning, it should leave enough space so he will be hungry in the morning and start day off good.
Other opinion is to start stretching this 5 am feed by example 15 minutes. As when he wakes at 5, you try to get him back to sleep for 15 minutes and feed only then and then next day you try to hold off 30 minutes and so on.