Speaking from experience, I tried out the Napfenyes restaurant. The fried banana and walnuts in caramel sauce was something I enjoyed. If you are looking for pocket-friendly restaurants with wholesome food, you can try MyCanteen near the city market.
Hungary is also famous for its Ruin Bars. My walking tour guide Susie had strongly suggested to try out a place called Szimpla. If you are looking for local specialty drinks, you might want to give Palinka a shot.
Here are a few tips I feel can help you make your experience (as a solo traveller or otherwise) a lot more enjoyable.
1. Try to read up a little about the different attractions and monuments, and probably a little bit of history - it helps connect better with the place, and makes the experience much more memorable.
2. I would usually start as early as I could in the morning and get back to the hostel by lunch time and rest for a couple of hours. After checking out a few more places in the evening I would return to the hostel again to catch on some rest.
3. Before you visit, try downloading apps such as Triposo, which gives you ALL the information you need about EVERYTHING in one place - right from the history, to food and the common phrases used in local language.
4. There are apps such as Viator, which help in booking experiences art reasonable rates. Do check it out.
I hope my post was of some help. Here are a few links to my blog, in case you want to know a little more about my journey :)
The River Cruise
The Walking Tour
The Danube Promenade
The Liberty Statue and Andrassy Avenue