Setting Up a JavaEE app on an Arduino Yun Board
Sorry, forget it. 64Mb ram, 16Mb flash what the Yun has.
Java EE is magnitudes bigger (and bugger) than that.

You might try with Java SE embedded, the smallest profile might fit to 11Mb:
http://www.oracle.com/us/technologies/java/oracle-javase-embedded-ds-365290.pdf
and then you can roll your own web server. 

(Personal comments: For little timesheets JavaEE is overkill, for anything modern JavaEE is obsolete - consider modern Java micro web frameworks, which offer magnitudes better performance than JavaEE).

What you can try is Raspberry PI, which works well with network, and Java (SE).
Not that well but also Java EE works: https://blogs.oracle.com/theaquarium/entry/glassfish_4_on_the_raspberry