You know that adhesive white foam draught-seal strip that you can get to seal the gaps round your doors...?

Yes, it can work very well if it's thick enough to seal the gap. But it can be difficult to remove. Why don't you tell your landlord that cold air is blowing in around the doors, you want to weatherstrip but want to know what he'd prefer to use? It's really the landlord's job to do this kind of stuff anyway. He should at least be paying for the materials.