I'm in the minority but I think it's totally doable!
Get yourself to a supermarket when you arrive and stock up the apartment and have most meals there. You can buy snacks at the supermarket and take these with you when out and about. You can buy an insulated flask each and take water out and about with you, keeping it cool for a while. You can buy a huge bottle of water from most shops near the beach etc to refill.
We went on a lovely holiday recently with a much smaller budget for spending money meaning we couldn't afford many activities at all. Luckily my child was very happy with the beach and pool each day! Can you check if there are any free places nearby such as sightseeing or museums/galleries? If you're taking a car (or would you consider renting?), you should be able to find a fair few that are accessible.
Big activities can be a treat a few times throughout the holiday and it looks as though this would be affordable.
It won't be a lavish holiday, but definitely affordable with some treats!
Sit down tonight and google places local to the apartment that you can get to by public transport that are free or very cheap. These can account for a few days of your holiday.
Google big activities that you definitely want to do and work out how much needs to be kept for this.
Google the local supermarket and set yourself a budget for 2 weeks of shopping.
Decide how much spending money you want to give the kids each day (there's plenty of time before you go for them to earn this).
Basically, just sit tonight and allocate your money to: food/snacks for the apartment, transport, spending money (toys etc for the kids), and places you want to see (free/cheap).
Then see how much is left over.
Is there enough there to eat out once or twice during the hol? (Google some local restaurants and check their menus for prices). Is there enough left to do one or two big activities? Is there enough left for extra spending money which can be used freely e.g. on extra drinks/snacks while you're out.
This is how I plan every holiday and it makes me feel much better knowing how much I have for each 'holiday area'.
You're definitely doing the right thing not putting any of this on a credit card by the way! I know it's tempting, but please don't start to go down path. I only ever use my credit card for absolute emergencies as it is so easy to see it as 'free money' and get trapped in debt.