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.
NTU/ Taida's library link, which may be useful to you: lib.ntu.edu.tw/ENGLISH/index_e.htm