Depends when you go and where from. Use Skyscanner with destination as everywhere to find cheapest routes for the days you want to go, use Google flights for price tracking and up to date prices.
Sign up to Secret Flying UK departures email mailing list for bargain alerts.
Cheapest cities in Western Europe are probably Porto and Lisbon.
Elsewhere, Poland (eg Krakow) is cheap for accommodation and eating out. I would guess Budapest would be too (have never been). Bulgaria is cheap, but Sofia perhaps not particularly appealing, Plovdiv maybe an option.
If you go in the shoulder season (May/June and September/October) you get cheaper accommodation, smaller crowds and still a good chance of decent weather.
Off season is even cheaper but less chance of good weather, although that's arguably less important for a city break.
Shoulder season you can get apartments in many cities for about £20 per person per night for 2 people.
Usually a better deal than hotels and you can self- cater to cut costs.
If you want to eat out, then make a lunch your main meal (as it's much cheaper than dinner) and then you only need a light dinner and can self-cater breakfast and dinners without doing much or any cooking. Or picnic and bottle of wine for sunset if weather is good.
Try Airbnb and also Booking dot com but try and book direct as commission charge is 15 to 20% at booking (charged to the provider not you, but obviously they have to factor that cost into their pricing).