Train? There's a novel idea :-)
There is a train that departs Vienna about 7:45 and takes about 2:40; so you arrive before 10:30am. There is an earlier one, but ....
The last train home leaves at 8:20pm. About the same travel time. The train arrives at Keleti station. In the center of the station towards the front are steps that go down to the lower level. At the bottom on the right you will see an information office. Buy a 24 hour travel card. The price is 1.650 forints (about $6). With that you are set to see the town. Without a lot of detail (which if you go through with this i can help you with):
From Keleti take the M2 one stop and then walk along the great loop road to the The New York Palace Cafe. Stop for coffee.
Back on the loop get on the 4/6 tram two stops to the Oktogon. Then walk up Andrassy ut to Hero's Square. If you get tired, the M1 metro runs directly under Andrassy and there are a number of stops along the way. But the old mansions and embassy buildings are worth the effort.
From Hero's square get on the M1 back in the direction you came from and get off at the Oktogon, then continue walking Andrassy ut in the same direction. Now its mostly high end shops and pretty nice architecture. If you get off at the stop before the Oktogon you can visit the House of Terror before walking down Andrassy ut. Horrible name, great museum to the evils of fascism. Plan on 2 hours if you stop.
At the end of Andrassy to your right is St. Stephen's Basilica; but we will save that for near last.
Walk the opposite direction from the Basilica towards the yellow trams (47/49 Tram) and hop on. Sit on left facing forward. When you see the Dohany Synagogue, get off at the next stop and walk back about a block. Take the short tour they offer.
Back on the 47/49 tram to the stop just before the bridge to Buda. To your left is the Great Market Hall (closed on Sunday)
Walk towards the river and see the stairs going down. At the bottom is the stop of the No. 2 Tram (now you understand the Travel Card.). The trams are like low cost HOHO buses, but a lot more fun. Take the tram going left to right. This follows the river front and the views are great so sit on the left. Get off at the second stop and walk away from the river about a block and you will find Vorosmarty ter which anchors one end of Vaci utca the famous shopping street. Want a snack? Go to Café Gerbeaud.
Back on the 2 Tram to the first stop behind the Parliament. First walk down the stairs to the river, cross the street and on your left are the "Shoes on the Danube". Then back to the rear of the Parliament. Now continue walking upstream past the parliament and up Falk Miska utca where all the basement antique shops are. I can give you the name of one that has endless connected basements and wonderful things to see. At the end of Falk Miska is the 4/6 tram line again. Take it to the stop just across the river in Buda. Then the 19/41 tram to the Chain Bridge and the Funicular up to the Buda Castle District. Use a guide book on castle hill. Figure nothing less than 2 hours up there.
Back down the Funicular (or walk) and then across the chain bridge to Pest again. Walk behind the Four Seasons Hotel and you will see the pedestrian area that leads back to the St. Stephen's Basilica. See the Basilica... and shop the pedestrian area. The best souvenir shops are to the right of the entrance to the Basilica.
Okay, now, before we go on. All those highlighted places are worth you researching. Decide on the house of Terror and if there is anything in particular you are interested in, like the old jewish ghetto area or parks or museums or views or communist / '56 revolution / WWII or wine or beer or bars or clubs or food or.... let me know and we will weave it in and see how much time is left.