From what I've read on the internet, it should be possible. But do keep in mind that anyone else that uses a PLA system like you will be able to also share your landlord's internet the same way. A PLA system basically shares the wiring that used to power the apartment complex, and if there are a bunch of others using PLA systems, then your speeds might still be very bad. After you install the powerline adapters, make sure you use the builtin encryption to keep others from gaining access to your network, there's a reason it exists.
Edit:
I found some good information here:
11. How far do powerline networks reach?
It's hard to find a good answer to this. One thing that is a given is that powerline networks won't work across transformers. This usually prevents your neighbor's powerline network from connecting to yours. But in apartment buildings there is definitely a chance of your network reaching beyond your unit. So you should take the steps outlined in Question 13 to ensure your network is private.
As for distance within your home, some sources quote 300 meters, while others say 200. NETGEAR's powerline FAQ takes an area approach and quotes "5000 square feet or less".
In general powerline networking should work in a typical home or apartment and might even work to provide Ethernet to a separate garage or outbuilding, if the distance isn't too far and it is on the same electric service.