Let's say you have a player who is 219 days old and the minimum is 220. Can you wait one day and sign the player a day into the season when he turns 220?

I think the cap ages too. So the player will age one day, and so will the cap.

the cap is a rolling cap. For every day in the season, +1 is added to the minimum and maximum. So the answer to your question is no, you could not do that.

Does this happen on the Fundamental league as well?

Kind of. Minimum will always be 0 there, but maximum will go up +1 each day.