I only managed to remove the fixed cup but the other one on the first picture I posted the other day is absolutely stuck. Do you think I'll be able to fit new bearings even without removing it?
It's possible. I assume by "fixed cup" you mean the right side one.
And you've pulled out the spindle, right?
You should be able to hook out the caged bearings the other side.
You need to get it properly clean, shove generous grease in and new caged bearings (could do loose but tricky given only one side access).
The left side one will be a lock nut (the hexagonal nut) and then the cup itself.
@Big John he speak sense v v v : get both cups off, measure the spindle length and get a UN-300 BB or similar (clean threads before installation obv, new BB will have threadlocker as supplied so no grease). You'll need a 20 spline tool but think BB and tool <£30.
I'd persevere in trying to remove both cups and replace with a sealed bottom bracket if it was me. You'll need another tool but they're not overly expensive. There'll be no adjusting to do. Fit and forget.
Suggest edit name of thread to 'Bottom Bracket service?"