Hi there, welcome to TSC and congrats for the record :) If you have read completition rules...
*When not using Time Attack mode, the difficulty setting must be Normal.
*Where available, the game's Time Attack mode is to be used for setting records.)
.. so if you have played the time on Time Attack and on Normal difficulty, you're good to go :)
To submit, you want to click on Submissions (left bar, big title) and from there, choose Sonic Advance and Time division. Hit OK and you'll be taken to submitting page, find the correct Stat and put your time there. When you're done, click submit from the bottom of the page and you'll be taken to comment page. If you want to attack your proving picture to the record, put the URL address of the picture to your comment. Then submit it, wait a moment and your Stat should show up in New Stats page :)
Picture is enough to prove as far as I know, so no worries :) Good luck to competition!!
EDIT: I mistakenly wrote SAdv2, so I edited it to SAdv1.