Yes, you definitely need your landlord's permission - that's not optional. Even if he's been reluctant so far, a heat pump is considered a structural alteration and requires his written consent. Without it, you can expect him to require you to remove it or potentially create problems when you get your deposit back. It's worth having a conversation and explaining to him that it also increases the value of his property. Some landlords are more flexible if you make a professional argument.
If he agrees: An air-to-air heat pump (heating only, not cooling) for a 75 sqm apartment will probably cost somewhere between €8,000 and €15,000 net - depending on the model and installation effort. On top of that there may be costs for piping and electrical modifications. That's obviously a decent amount of money, and whether it pays for itself really depends on how much longer you'll be in the apartment and what your current heating costs look like.
Honestly, if the landlord won't cooperate, I'd rather try heating less - better seals on the windows, turn thermostats down, ventilate strategically. That costs much less and works without permission too. A heat pump makes more financial sense in a house you own, where you benefit from lower heating costs long-term.