Fix water pooling on porch connected to driveway
Yes, you could try the asphalt patch but after a few winter freezes, it would crack and crumble.
A better, more permanent solution would be to dig out about 12 square feet in front of your stairs, about a 3.5' x 3.5' area. Regrade it to eliminate the pooling and install patio stones/pavers. You'd loose some asphalt but it's not that great to look at and you'd have a nice landing at the base of the stairs.