Hey there!
Definitely eat in the Gothic Quarter, lots of cheap tapas bars that won't set you back.
I would recommend the same when going out, the clubs near Barceloneta beach, while they are impressive, are really expensive to not only get into, but drink at well. Many bars dotted around the Gothic quarter (particularly towards the beach end) have extremely reasonable price.
Hostels shouldn't be an issue, there are tons in Barcelona that are very well priced.
When it comes to attractions, if you are to prioritise:
Go inside the Sagrada Familia (I didn't do it when I was in Barcelona and I am still kicking myself for it)
Go up to Park Guell (it's free!)
Go up to the top of Mont Juic (also free - the views are awesome, has the Olympic grounds including the best view from a swimming pool in all of Europe)
Relax on Barcenoleta beach - just watch your things!
Walk the city - many of the highlights of Barcelona can be within walking distance so make sure you book a centrally located hostel.
That's all I can think of right now, I hope this helps!
Matthew