Assuming the train from Rome to Prague passes through mountains or valleys flanked by mountains at some point between northern Italy and Germany (i.e., the Alps), is the scenery by train worth a day trip rather than taking the night train? We would be giving up one day of only three we have time for in Prague.
Several airlines fly non-stop taking two hours, costing you only a half day. www.skyscanner.net
On your very tight schedule you face a choice between scenery and city touring. Because there is competition, rates are budget if bought well in advance. This advice is less true for the Venice leg of your tour, also posted on this forum.
Night train from Rome goes to Munich then you have to change to DB bus which takes you to Prague.
This is 16 to 22 hour train trip. Flying is the way to go.