How does one distinguish between homographs in Chinese? How does one distinguish between, for example, 便宜 (cheap) and 便宜 (convenient) in written Chinese?

The context may give no help whatsoever, cf.


  我的大学很便宜。


(Bad example, please excuse my awful Chinese. Feel free to edit with a better one.)

Is there a general rule for identifying which meaning is intended when there are identical characters?
Yes, this word has two different pronunciations.
便宜 pián yi means:

[adj] cheap, inexpensive
[noun] benifit, interest, which you are not supposed(allowed) to get.
[verb] to make sb get that benifit

便宜 biànyí means:

[adj]convenient

Because these meanings are different significantly, I don't think we'll get confused. Also, biànyí is a written, literature ,classic word, so you won't meet it in most cases. Instead, we'll use &quot;方便&quot; in that case.
In your example, I don't think 便宜 is used correctly, no matter how it pronounces. You may want to say:

我的大学学费很便宜。 The fee of my university is very cheap.
我的大学上网很方便。It's very convenient to surf the web in my university.