Will be in Florence for one day with my son. Would like to climb to the top of the Duomo, can't find info via Google to confirm if the stairs are open on Sunday.
Dome open Monday to Friday 8.30am to 6.20pm; Saturday 8.30am to 5.00pm. Closed on Sunday and on Catholic and civil festivities. Rites in the Cathedral may bring additional closures or change of times.
Just the bell tower or the top of the entire church? Rick Steve's tip was spot-on. Go to the Duomo museum and sign up for the tour of the Duomo. It's relatively inexpensive and you get a set appointment that allows you to bypass the long line for the bell tower. It's really a great tour and you get to travel along the top of the church (inside and out).
There are two possible climbs, the bell tower or the Duomo. Bell tower may be open on Sunday (separate buildings).