What are the internet options for Kuwait? What about getting an international hotspot like Skyroam?

Many options for internet in Kuwait.

Free wifi on posts like Arifjan at the Starbucks, MWR in Zone 1, USO in Zone 6.
The best options on post are the Qnet tied to a wifi, which you can purchase via a website, the information is at housing or main PX on Arifjan or Buehring. If you want to travel and have coverage, you can purchase a wifi hotspot, and a SIM card with Ooredoo (Arifjan or Buehring) or Zain (off post or KNB).
If you are in housing and have the wifi, the 1.5Meg is the cheapest per month option at $35.
I have used Zain for when I travel and you can by 100G for a 28 day cycle for $60.
I have used Qnet and Zain and find them both perform well at a reasonable cost.

@ssg Christina Barnes SFC (Join to see) and @LTC John G Shaw both mention great options. I was able to use my PCS orders to Kuwait to get AT&T to unlock my iPhone. Then I just purchased a Kuwait SIM with a data plan. I used the hotspot feature to power my laptop when I was in the barracks and still had internet and phone use when around post and not near the free wifi spots. It was about $45-$55 a month for 2-3GB (in 2014). I wasn't streaming movies in the room much but it was good enough for most my needs. Plus we had DSN lines in the barracks so I usually used that to make morale calls back home. Worst part of being in Kuwait: being away from my wife and kids. Best part: cheapest and best haircuts since my days on Fort Sill! LOL But seriously, it wasn't that bad of a tour. I got into great physical shape while I was there. Just wished I had focused on college a bit more.

Theres only 3 sim card providers here. Ooredoo, Viva, and Zain. Zain tends to have the best service that I have seen. Viva is the next best, and Ooredoo is the worst, but they are pretty much the only service that is sold here on Camp Buehring.

There are craigslist groups that people are always selling sim cards and pucks on. I would start checking those locations if you are looking for one now.