It is not your overflow it's your landlord's overflow.
It should be relatively easy to fix but if you attempt it yourself and mess it up your landlord would be within his rights to charge any repair bill against you.
If you call in a plumber yourself then you will have to pay him unless you have the landlord's agreement to reimburse you.
The people with the problem are your downstairs neighbours - get them to worry the landlord.
Peccavi, April 2009