The International Herald Tribune is the international version of the NYT. You can find it at most bookstores. Some cafes subscribe to it. You can subscribe to it, too. It's not the full paper, but it has highlights from the regular print edition.
There's also a kickass bookstore on Dun Hua Ln 63 (near intersection with SiWei St) that rents international magazines (mostly from UK, US, and Japan) along with a good selection of domestic mags. It used to be one of those big chain movie/comic rental shops but it went independent a year or so ago. I'm still looking for a store like this in Taichung.
You can get the actual NYT flown in from New York via a few different courier services, but the service costs an arm and a leg. You can subscribe to Economist or any other mag through the publisher; international subscriptions are about twice the price of the regular subscription.
I've never seen the NYT for sale in Taipei, and that Sunday edition is one of the things I rejoice in on visits home (and reading it leaves me no time for a newspaper for the remainder of the week!).
As others have said, the IHT is available at eslite bookstores (and I think Page One as well), if you want to read the same stories you saw online a couple of days earlier. They have a Saturday/ Sunday weekend edition that is far far thinner than the Sunday New York Times.
Kindle monthly subscription to the NYT is $28 (US); the Saturday issue available right now is $2 - I think the Sunday edition is the same price. There are numerous complaints on Amazon about the Kindle edition missing substantial amounts of content, but I can't speak to that. If you want the Sunday NYT, the Kindle edition is almost definitely your best bet.
My own university has the Washington Post, not the NYT, BUT it's surface delivered so it arrives several weeks after publication. I don't know if that's typical. (And I really wonder what the point is.)
The other magazines you mention may show up in university libraries: if I'm searching Taida's library correctly, they've got Cosmopolitan and Esquire, but not Vanity Fair (mind you, that Thackeray novel makes "Vanity Fair" searches tricky). But will you have access to them? Normally university libraries are reserved for students and staff only. I've heard the public libraries are OK, but I've never used them.
For magazine articles in general: do you know about Instapaper? It's a great program/ app that lets you save articles to read later. I use it all the time for longer articles that I find online, then read them on my iPhone before bed or on the MRT or whatever.
I don't have kindle or iPhone. I have no intention of getting an e-Reader. May get an iPhone before getting on the plane.
I wont be a student at NTU. But I will have access to NTNU's library since I'm going to the Mandarin Training Center. I'm pretty sure just the fact that I'm a student will allow me inter-uni-library access around the island. I can't imagine that would be a problem.
Several week delivery huh? We could go through 4 civil wars, 3 terrorist attacks, 2 assassinations, 1 financial crisis, and the opening of North Korea in that time period.
At my own university I need to "sign out" an NTU library card (a week for me as a teacher I think; less for students) that can then let me into that other library, so I need to plan a little bit. I don't think you can just use your student card at any university library, but with a bit of planning, yes, you probably can get into the other libraries.
Slightly OT: are American iPhones still SIM-locked? If you do intend to buy an iPhone (big if, I know), then I suspect you'll be better off getting one here.