There are times when I (instantly) fall asleep (instantly).

[Rachel Adams]

Can I use "instantly" before "fall"?

There are times when I (instantly) fall asleep instantly.

 

[5jj]

Yes, though it's not as natural as it is in the end position.

 

[Barque]

I'd prefer "in an instant" or "immediately" or "as soon as I lie down/my head hits the pillow", but maybe that's just me.

 

[Tarheel]

Can I use "instantly" before "fall asleep"?

There are times when I (instantly) fall asleep instantly.

I might say that there are times when I fall asleep right away.