<ol>
<li>How do I get a method's execution time?  </li>
<li>Is there a <code>Timer</code> utility class for things like timing how long a task takes, etc?  </li>
</ol>

<p>Most of the searches on Google return results for timers that schedule threads and tasks, which is not what I want.</p>

This probably isn't what you wanted me to say, but this is a good use of AOP. Whip an proxy interceptor around your method, and do the timing in there.
The what, why and how of AOP is rather beyond the scope of this answer, sadly, but that's how I'd likely do it.
Edit:
Here's a link
to Spring AOP to get you started, if you're keen. This is the most accessible implementation of AOP that Iive come across for java.
Also, given everyone else's very simple suggestions, I should add that AOP is for when you don't want stuff like timing to invade your code. But in many cases, that sort of simple and easy approach is fine.