Does this mean that he will remind me about this moment in the future, or he will remind me (of something unnamed) immediately?
The former. את הרגע הזה is the direct object.
Thanks. That was my first thought, but then I was not sure whether ת הרגע הזה might not be a sort of fossilized expression of time, as most languages have.
Hu yazkir li et ha'rega ha'zeh
Is he talking about an inanimate object?