Hmmm - well at a year old, I personally wouldn't let him play. His jaw isn't really strong enough or properly set yet. At that age, I'd be careful about how strongly he played tug...
If you do though, how long is a matter of judgement. As an arbitrary measure I'd say half an hour is enough, but it would really depend on just how physical he was being, or if you detect any signs of tiredness that excitement or determination isn't letting him give in to.