Ciao Barcelona,
May I ask where you have read that the Uffizi and Accademia will for sure have
free entry on April 1st? I have yet to see any official confirmation about it. They will be open for sure, since they are always open on Easter but whether free or not has not been confirmed.
One can hope they will honor the "free first Sunday of the month" this year but Easter Sunday is a very profitable day for them, very very busy day and booking tickets is the main way they manage the amount of visitors on that day.
On the free Sundays, the thing is you
CANNOT BOOK tickets ahead of time so I see a nightmare for visitors if they honor the free entrance that day. It means LONG LONG lines and waiting times. You do have to wait in line and get a ticket before visitng the museum.
The ONLY way I know to skip lines on the free Sundays of the month is to buy the
Firenze Card but at 72 euro per card if you're only in Florence on that Sunday going to the Uffizi and Accademia would be costly. If you're here for more days, you should consider it. Make sure to read this:
http://www.visitflorence.com/florence-museums/is-firenze-card-worth-it.html
I am not saying anything on whether April 1st will be free or not, just wary because I have not seen anything confirming it (and I keep a look out for these types of news

). I can only say that the city of Florence, which manages other museums in Florence such as Palazzo Vecchio, has moved their "free first Sunday" (which is only for residents) from April 1st to the 8th, for that precise reason.
As for Easter Monday: both museums are usually closed on Mondays but the last 5 years they have done a
special opening on Easter Monday to open the museum up to the crowds. So while we can safely assume it will be so again this year, you really cannot say it without having official confirmation from the museums. Unfortunately these types of official news only are given out a few weeks beforehand. If you want to consider getting the Firenze Card, just wait and get it here, that way you have time to see whether the museum will be free or not and to book tickets or get the card.