The document RC1981 referenced is a great document, fantastic really, unfortunately the historic district they describe doesn't extend to where 305 is.
The present low-rise apartment building on that spot is now known as 303 E 71st St. The next building is 311 which suggests that there was perhaps more than one building on the site of the present 303 (such as "305").
If the online records don't go back far enough, which appears to be the case here, then it is time for a trip to wherever they keep the records. (Of course, if it's a famous building, a google search of the address might reveal good info.)
First there is the Certificate of Occupancy:
Department of Buildings
In the bottom left under "Buildings Information", fill in the info and submit. On the next page if you click on "View Certificates of Occupancy" you will see it only goes back to 1947, which is the present building, so that doesn't help with what was there in 1918. I've never visited the Dept. of Buildings so I don't know what kind of info they have publicly available at their offices. But from its name it would seem like a good place to start.
The online property records is also a useful website:
ACRIS: Online City Register
Click on "Begin using ARCIS"
Click "Find Addresses and Parcels"
Select Manhattan from the drop down menu
Put 303 0r 305 in the street number box (either search reveals the same documents)
Put E 71 ST in the street name box
Click "Find BBL"
Click "Document Search by BBL"
Click "Search"
ARCIS only goes back to 1966, so again that doesn't really help here. You'd have to visit the City Register to go back further. Now this often won't tell you the type of building, just who owned it, but other documents that have been recorded on the property can give clues to what was there.