To access online mode you need to go to the DigiLab and interact with the red circle in the back of the room. You WILL need PS Plus for this.
Titles are unlocked by progressing the Offline Colosseum, buying them in the online mode coin shop, and finishing battles online in Event mode.
You automatically start with 3 titles and after doing all the Offline Colosseum cups you'll gain an extra 6. The final title can be obtained in any way mentioned above and the trophy will unlock once you've purchased or unlocked your 10th.
If you choose to buy any titles (100 coins each) and you just want the trophy you can buy them, get the trophy, and reload your game as long as you don't save as the coins are tied to your save file and NOT your account.