Here is the Illustrated Parts List:
http://bsintek.basco.com/BriggsDocumentDisplay/Z6jmyBVJ1DajI.pdf

Here is the Operator's Manual:
http://bsintek.basco.com/BriggsDocumentDisplay/flivAJX-nfBhU7y.pdf

Your description of the symptoms sounds as if you have not been using the choke, which is circled in yellow in this picture:
[Linked Image]

Pull the choke all the way out (i.e. to the left in the picture - it will come out about 10 mm). When you pull the starter it will start, probably on the second pull after months in the shed, then after about 5 seconds it will stop if you haven't been able to push the choke in by then (I never can.) If it has stopped, push in the choke and it will then start first pull.