Good luck to you and to us all!

The best way to go about it from my experience:
1. optimize, repair tables and backup the db as it is in it's current form.
2. Disable all the extensions. make another backup and name it something like --noext
3. Do a clean install of the new phpbb 3.3.x IN A NEW DIRECTORY (because you still have /images, right?) This will prevent you from overwriting stuff. Get a URL for the new directory.
4. create a new DB and import the tables and data to it.
5. goto the config.php and change the db name and the user/pass.
6. run the update from the /install directory on the new URL

The key is to remove any extensions (disable) and that way you will minimize the issues, maybe perhaps not eliminate them entirely, but it will get most of the problems sorted.

Your theme will not survive, and the site logo batter etc will evaporate because of the changes. even between 3.2.x and 3.3 it's a nightmare

If you're on 3.0/3.1 etc then follow the multi-version-staggered update on phpbb.com

I really hated 3.1.8 to 3.2.2 and then multiple 3.2 updates were half-ok. then 3.2.8 to 3.3 was another massive nightmare.

If you can do the update on an offline instance (localhost) for the install and DB and then upload the result, it may be easier.
just make sure you have this one running and live (admin disabled board) so you can use it as a test reference.

In my experience, despite the phpbb claims, it's not ever a straightforward update. There is always something they forget or stuff up and it takes about 3 updates to make it work.

and some very good extensions are no longer functional on 3.3, which annoys me no end. Especially stopforumspam and other extensions.

Best of luck. I know you're going to have a bit of a rough time ahead. take it easy, don't rush it.