There’s a sufficient number of good students who’ve had Latin for it to be virtually a non-factor in admissions. The Ivies and peer schools draw heavily from private schools and top publics, many of which offer Latin. In any case, East Asian studies departments need students too, and they’re often not much bigger than classics departments.
I recommend Chinese. It’s a very useful language today, and as DeepBlue86 mentioned, it can take a few years to get to a good level of Chinese.
If you really want to learn Latin, you could teach yourself by using Wheelock’s excellent grammar; as ancient languages go, Latin is certainly one of the easier ones. Working through the grammar can take about a year if one moves slowly or a few months at a fast pace. There’s some excellent resources online, such as the [Dickinson commentaries](http://dcc.dickinson.edu), and even smartphone apps for practicing Latin grammar and vocabulary.
Latin literature (both Roman and medieval) is definitely interesting, but Chinese has a long and rich history of literature in its own right.