Welcome to ESOUI!
You should read these two things first if you are just getting started:
The wiki also contains plenty other useful resources and if you have questions you can always pay us a visit in the
ESOUI Gitter channel.
Making a patch for an existing addon is basically grabbing the addon, modifying it and uploading it as a new version of the addon. If the original author has gone missing and you plan to take it over, you can try to contact them via pm or reach out to Cairenn or Dolby (admins) and ask for advice.
Testing an addon is only really possible in ESO itself, so you will need to have it running. Its Lua engine is based on Lua 5.1 so you could try run some unit tests, but you will have to add a lot of mock objects in all likelyhood.