After all this you will have, what, 9k left to save or play with ... 13000 gross is indeed a huge wad of cash in Lodz.
Not quite. The amount is gross, so quite a lot of tax and insurance will be taken off before he gets the money. He'll take home less than 8 or 9k
Remember, a job in Poland that pays that amount will probably involve a certain quality of life. Certainly not turning up to work in the same cheap suit day in day out, skimping on taxis, living in a tatty flat, skimping on vacations etc. If someone's on a good wage, they usually expect to live well. I'm on much more than his gross amount even after taxes and certainly don't feel either rich or that I could live like a student or a miser just for the sake of skimping. There have been times in my life when I've had not much more than 150zl to spend on food each week and I certainly don't intend to do that again just for the sake of it.
Nevertheless, €3000 gross in £ódź isn't bad at all, and the OP should be able to have a decent lifestyle and save some money, especially if he is single.
By the way, I eat out (at moderate restaurants) several times a week and can still cook very, very well indeed ;-)