Slowly might be the best approach ... especially via git and using the scripts in moodlecode/admin/cli/
Just one more piece of advice ... at each successful step ... like from 2.2. to the 2.3 ... do a full site backup of the site ... 2.3 code directory, 2.3 data directory, and an sql dump of the 23 DB. Yeah, I know ... could be lots of space and takes longer, but that backup becomes a 'fall back' if the next step up creates real issues which will take time to resolve (ie, ELIS issues).
Besides that, one will get plenty of practice at upgrading via git. After awhile, server OP just might decide to bash/shell script it and then what a breeze it will be on the next upgrade. ;)
The 'march' is somewhat a pain, but completing the journey is worth it ... IMHO!
'spirit of sharing', Ken