From what you've said - go. You need to have a real cheapskate mindset, and be honest to your kids - there's no money for anything you have to pay for (or else there is £5 a day total or whatever) so they will need to think about exploring, looking at a new city, finding things out, making their own fun on a beach you can walk or get a bus/train to.
My mum took us abroad sometimes on a real skint budget and although mostly we camped in the UK, we did manage 2 trips to France when ds was small, so I've got some idea how to do this.
You're going to have to say 'no' a lot, but cheerfully. You're probably used to that. Kids are used to it too. Don't stress about it.
You are going to need OK shoes for everyone - lots of walking. Make sure they're not grown out of them just as you leave.
Airports are money sponges these days.
Take sandwiches to prevent having to buy food there.
Research public transport from the airport.
Set a destination per day - don't think 'not everything can be free', think 'we need to find free things to go to'. Get guidebooks from the library. Choose one thing to go to each day. At least 2 beach days.
Even/especially supermarkets when you're abroad can be brilliant fun, I still like a supermarket on holiday now. Looking at all the different brands and flavours, take your time, and they're usually air-conditioned. If you can afford to come out with a packet of sweets or biscuits or juice each time, so much the better - maybe one child chooses in turn each day.
Think about some cheap things to do in the apartment. Get new childrens books from.the library. Mum made us write and illustrate holiday diaries with paper from her work she'd stapled together. Put the pieces of a biggish jigsaw in a sandwich bag knotted tight, and take a picture of the jigsaw on your phone, and have that going all week. Would your children be up for doing Portuguese on duolingo? We used to get audio books from.the library again but that stuff is easier online now.
Are your children old enough to earn a little bit of their own ice cream/treat money? Even if they have to 'earn' it from grandparents etc? Make fairycakes, sell some bits, learn to crochet mice or something?