I have attended at a local Parish nearby for perpetual adoration for one hour. Usually it is simply just a quiet hour of prayer and devotion before our Lord in the Blessed Sacrament. The Consecrated Host will be placed inside the Monstrance, to be viewed and worshipped by all, by the priest or sometimes a deacon.
Something I like to do is bring a small Bible and pray the Psalms while kneeling in front of the Eucharist. This is an awesome way to pray and worship our Lord anyways, but any prayer is especially powerful in front of the Eucharist.
You can always simply meditate on Scripture, or pray a devotion of some kind like the Rosary or Divine Mercy Chaplet, or simply just kneel and take in the awesome presence of Jesus.
Parishes that have a designated time each week usually I believe you can stay for the full time if you wish or you can stay simply as long as you can. Definitely be sure to be quiet and respectful of those around you though if you do need to get up and leave before the time is over. I would recommend staying the full time of adoration though because it really is that awesome and powerful to me.