I would go along with Steve. Te capacitor is there to give the motor a 'kick', for want of a better word, to get it to start and if it is running OK with a mechanical kick from you then that's almost certainly what it is.
If you aren't sure how to do it then get a friendly electrician to give you a hand. It shouldn't be an expensive fix and better than you possibly screwing it up? Sorry if this is egg sucking and granny?