Located right in the city center of Budapest in a building that used to be a 4-star hotel is Wombat's The City Hostel Budapest. You'll be able to get anywhere in the city easily from the Deak Ferenc Ter Station that is close by. All of the train lines make a stop at this location.
The hostel puts on a breakfast buffet every morning that you can pay a surcharge to partake in. You'll find a common area that has both a refrigerator and a microwave for your convenience. There are also laundry facilities on the premises and TVs in multiple shared living spaces.
Wombat's The City Hostel Budapest has 40 rooms at the property. They're dormitory-style and you will be in a room of mixed travelers where there are bunk beds for guests to sleep in. All guests will have a locker provided to them. The bathrooms are shared with the other travelers you're bunking with and include only showers. Your bed linens will be included and if you need towels, those can be requested from the front desk for free.
There is a bar within Wombat's The City Hostel Budapest, but there isn't an on-site restaurant. This part of Budapest has over 400 restaurants that you can easily walk to, though. Stand Restaurant is right outside the hostel property. Daniel's Soups and Sandwiches is, as well.
St. Stephen's Basilica is only seven minutes away on foot. Buda Castle is a little further away but within walking distance still, as are the Rudas Baths.