So a couple of things here...
Regarding hospital stays: Most Gastric Sleeve patients stay only one night in the hospital, but more and more, we're seeing this surgery being done on an outpatient basis. As a general rule, people recover better at home than they do in a hospital, so there's a really good reason to send them home as soon as possible. Gastric Bypass patients tend to stay in the hospital a couple of days, on average. It's just a more involved surgery and the risks of infection, dehydration, etc. are higher.
Regarding the question of pain meds: Generally NSAIDS are a no-no after any gastric surgery due to the potential to irritate the lining of your stomach. This pretty much leaves either Tylenol or opioids like hydrocodone or oxycodone. The problem with opioids is that they have a really bad side effect for most people in that they are quite constipating. Because constipation is a really common problem post gastric surgery to begin with, prescribing opioids is just going to make a bad problem much worse. This is not to say that surgeons will never prescribe them, they certainly will if you really need them, but it's never going to be the first line pain med post gastric surgery.
Bottom line, you'll probably be fine with just Tylenol, but if you really feel like you need something more, your surgeon may be willing to prescribe something stronger. Just be sure you know what you're in for if you go this route, because you may find the pain from opioid induced constipation is worse than the surgical pain!