Generally, you pay by the half hour or by the hour. Ask about membership once you've found a place you like because most establishments offer discounts to members. My favorite place, which is located less than a minute walking distance from my school, costs about 1,000 won per hour for membership. It's the kind that makes you wish you had brought ear plugs.
Most game room employees do not speak any English, so if you have any internet problems, you usually have to puzzle them out yourself. Most cities have an abundance of game rooms, so just shop around until you find one that you like. Think about location, air quality, noise level, price, number of computers, and shop hours.
These internet game rooms can do a lot to combat homesickness. You can keep up with news from your hometown by looking up the local newspaper. Urge all your friends and relaties to get connected if they aren't already. They can keep you updated on more personal news as well as provide a listening ear for your excited discoveries and heartaches.
You can also start your own website to chronicle your adventures in Korea, share information with other ESL teachers in Korea or advise others who are considering coming here to teach.