Get up early so you can go to the Parliament building and get tickets to go inside. Usually tickets are all gone before 11 am. Hope over to St. Stephen's Cathedral nearby and make sure you pay to go the top, the view is definitely worth the hike! The easiest way to get around the city is on the Mol Bubi Bikes, which are available to pick and dock next to every major monument.
Ride down along the Danube and cross the Chain Link Bridge to get to Buda Hill. Skip the Fenicular and just walk up. After, go to the Fisherman's Bastion, have an Aperol Spritz at the cafe underneath the Bastion, but get the coffee and cake at Ruszwurm Confectionary. The oldest operating cafe in the city, it still maintains all its charm.
If you walk directly downhill from the Fisherman's Bastion, you can catch the tram a couple stops (3 to 5 depending on where you got on) and get off at Szent Gellert and walk across the bridge to Central Market Hall. Get some paprika sausage and some goulash soup bases. Fruit and vegetables are also very fresh here.
New York Palace Cafe is beautiful, but overrated in my perspective. Worth it if you have an hour or two to kill, but Ruszwurm's desserts are still way more authentic.
If it's before 6 (it stays open until 10, you just have get in before 6), you can end the day soaking in the thermal baths at Szechenyi. If you're lucky you may catch the old man who just beats everyone at chess!