Just running sketches couldn't be easier. Processing 1.5.1 allows sketches to be exported as applications:
After hitting that, you should see the Sketch's jar in tie .pde's directory. scp that up to bug using your favorite scp client, and just:
java -jar Sketch.jar.
Here's the pointillism example (in processing-1.5.1/modes/java/examples/Basics/Image/Pointillism/):
Running the IDE on BUG is pretty easy too, though I'm not sure why you'd want to...
opkg update; opkg install openjdk-6-jdk
tar xzvf processing-1.5.1-linux.tgz
rm -fr java
ln -s /usr/lib/jvm/java-6-openjdk/ java
After that, just ./processing and close the dialog regarding openjdk-6. I'd suggest using it with our video module, as 320x240 isn't ideal for writing processing sketches ;D