Which grandstand(s) are you looking at? I'm reasonably certain that most, if not all grandstands areas have at least one screen.
I've been to Monza many times, the last time in 2019 (before covid) so I assume nothing drastic has changed since then :-)
Staying in Milan is the obvious choice for most people, as there are trains direct to Monza station, where a shuttle bus will take you to the circuit.
Bergamo is also a nice place to stay and also has trains direct to Monza station. Check TrenItalia for time schedules, you can buy your tickets at the train station ticket machines on the day
You should allow around 3 hours from the point where the train leaves the station, to the time you want to be sitting in your grandstand seat. There are often long queues for the security checks at the circuit, as they check everyone's bags. Usually there's a list on the circuit website of forbidden items, but I can't seem to find it right now. Maybe it will be updated closer to the race date. But basically food is fine, but many drinks (cans/bottles) are not. I suggest if you want to take your own drink, get a hydration pack (the refillable bags that you can wear on your back), at least in 2019 they were allowed in. I took two, one full of wine and one full of juice 
Fans are allowed onto the circuit after the race, so if you want to walk the track, allow some extra time after the race into your travel plans.
