This is the text:
ああ 僕 まだ 名乗ってませんね｡ あした お会いしたときに｡
It is referring to tomorrow, so I would naturally say: お会いするときに instead.
Why does he use a past form?
There is not any past form, this is the accomplished form.
"あした お会いしたときに" could be translated as: "once I met him/her tomorrow".
So, even it this takes place in the future, the person is speaking about when the action of meeting the other person is accomplished.
Whereas お会いするときに is the time where the person is about to meet (this is not done yet) the other person.