My favorite little engine. I would test the armature with a ohm meter, should be between 1 and 2 ohms to each segment. Make sure you have good brush spring tenshion on both brushes, and a clean commentator and all electrical connections are soldered good. Sounds like you have to spin it to get it started..
Also take a ohm reading on the field, should read about 1ohm or a bit more.
I agree with Marty. Offhand, it sounds like a faulty armature segment. That is, once rotation is started, it will bypass the offending segment and momentum will make it run. I had a 300 series engine that had a broken wire to the commutator plate, and once soldered, it ran fine. Armatures are out there as replacements.
Those 300 series locomotives are usually like the Energizer Bunny. Let us know if we can help, that's what we do!