Hope you get this.
If you get there early on the 10th, I'd start out early and go to the Great Market first. (It's just by the Liberty Bridge.) The Market closes early on Saturday and is closed entirely on Sunday. (Actually, I'd stop in one of the Metro Stations and get a 3-day travel card first.) I'd then take a tram from the front of the Great Market to Deak ter, and go to Erzebet ter to catch the #16 bus to the Castle district. Get off at the Matthias Templom and visit Fisherman's Bastion. (Great views of Parliament and the city from there.) Then amble down to the Castle.
There are multiple ways of getting back to Pest.
1. Catch the #16 bus at Disc ter. (Exit the Castle on the back side and follow the cobbled path up to Disc ter.) Leave the bus at Clark Adam ter (1st stop after passing through the tunnel and before crossing the Chain Bridge. (Note that you'll have crossed the Chain Bridge on your way up to the Castle District.)
2. Take the funicular down from the Palace to Clark Adam ter.
3. Walk down the steps to Clark Adam ter.
From Clark Adam ter, you can walk down to the Chain Bridge and cross it on foot, if you choose. ON the East bank, you can catch the #2 tram or the #16 bus (again) to take you back to Deak ter. Alternatively, You can either take a bus (I've forgotten the number) that takes you to Szent Gellert ter, or (with a short walk to the entrance of the Chain Bridge), a tram that also takes you to Szent Gellert ter. Szent Gellert ter is on the Buda side of the Freedom Bridge.
Note: If you aren't that early, I'd skip the Great Market, and go directly to Deak ter and to the #16 bus. to the Castle District. Either way, you'll 'get' the Freedom Bridge. (On the #2 tram just get off at Fovam ter.)
From Szent Gellert ter, take either the #47 or #49 tram back to Deak ter. Then take the Yellow Metro to Heroes Square. ( Hosok ter.) The Yellow Metro shouldn't be missed, incidentally.
Then amble down Andrassy as far as you wish. The Opera House isn't far. (I suspect that that may be what you mean by the National Theatre? If so, then the tours are limited.)
All that is indubitably too much for one day, but it might give you and idea of how to get around to your 'must see' sites.
I think that the only things I'm missing so far are Parliament and Szent Istvan's Basilica. If you want to see the interior of the Parliament, then advance tickets are pretty necessary. If it's the exterior, then the best view are definitely from the Buda side (Fisherman's Bastion and the Palace.) as there is currently a lot of construction going on in the area. I don't know whether tours go on on the weekends.
Szent Istvan's is easily visible from Deak ter and close to it. I suspect that Deak ter will be your 'hub' for transport, so fit it in when it's convenient. D o remember that the 11th is a Sunday, so there will be a lot of services going on.
I'd also add one of the baths as something one should try for. The most popular one is the Szchenyi, another stop farther from Heroes Square on the Yellow Metro line. Another one is at Szent Gellert ter. on either the #47 or #49 tram from Deak ter.