I've explored loads of European cities (very cheaply) on my own. I usually just do day trips but I've also stayed for a few nights sometimes.
I'd definitely recommend looking on the websites of the budget airlines for destinations that have got early flights out and late ones back (so that you can get the most of your limited time). As an example, you can fly out from Stansted to Madrid on 12 June at either 0625 or 0825 for £29.57 (arriving at 1000 or 1200). You could then return two days later (14 June), leaving at 1800 (arr 1930) for £21.99. Those flights would get you there in time for lunch on Day 1, with the afternoon and evening to explore the city. You'd then have all of Day 2 there, plus the morning of Day 3 (heading off to the airport after lunch).
You'd also need accommodation. This is my favourite site. (I've used it myself for accommodation in Madrid):
https://www.hostelworld.com/
Don't be put off by the word 'hostel' in that URL. There are loads of cheap hotels and B&Bs on there too. Just carry out an initial search and then use the filters to look for the type of accommodation that you want.
(Booking.com is worth a look too).
Using my example dates (and the Hostelworld website), you could stay in La Casa de la Plaza for £18.34 per night.
A 3-day 'T-dia' ticket, for unlimited travel on the Madrid Metro trains (as well as buses and trams), including travel to and from the airport, costs €35.40 (about £31).
So that's a 3-day (2-night) break in Madrid, with flights, accommodation and all local travel for about £120. If you eat in places like the very good "All U Can Eat" chain (where there's a wide range of food on offer, with coffee or soft drinks included, for €9.95) you needn't spend too much on food either.
That example took me less time to research than it did for me to type this post, so it's not hard to find a bargain. [If you need help, please let me the know the airport(s) that would suit you best].
Happy travelling!
(PS: For cheap travel insurance, try here:
http://www.holidaysafe.co.uk/ . I've just renewed my annual, multi-trip European cover with them for £15.91)