"I will remember this time."
Translation:Я буду пам'ятати цей час.
Do you mean like in "This time I will remember when your mother's birthday is"? Than it should be "Я запам’ятаю цього разу".
It is also а future form of пам’ятати (to remember), but буду пам’ятати it is like a pledge that you won't forget it, запам’ятаю - well, you will make efforts to remember it, but..