Hi,
The best option for you would be flying with a budget airline as they offer the cheapest prices.
The are however no direct flights from Newcastle to Bratislava. Your best option would be to take a flight from either Edinburgh, Birmingham, Manchester or London (
http://www.low-cost-airline-guide.com/en… You would have to book a flight (or go by car) to one of those cities and take a direct flight from there to Bratislava.
It might seem like a lot to do, but it is worth the money as traveling with budget airlines is much cheaper than with traditional airlines.
As for the hotels, I would suggest taking something once you are already in Bratislava. There are plenty of cute, small hotels (guest houses) in Bratislava so you will have no problems finding a place to stay.
Bratislava is a lovely city, and I especially like the old town which is very quiet and relaxing.
Have fun