How can I connect my 30 amp electric dryer to a 50 amp plug?
You have an incorrect receptacle installed at your apartment. Some genius just replaced the existing 30A dryer receptacle with the incorrect 50A range receptacle for some reason rather than doing it right. 

You only real option is to get the landlord/super/etc to have the receptacle replaced with the proper 30A dryer receptacle. The cord you have on the dryer is correct, albeit the old style 3-prong version.