How many numbers are there between 99 and 1000 such that the digit 8 occupies the unit's place?

To be precise you have asked for 3 digit numbers with unit place 8 (as 99 and 1000 don't have any)
_ _ _

we have three places to fill
_ _ 8

If we stick an 8 at the unit place we can still Fill second place using all 10 digits available (0 to 9)
Hence ways to fill 2nd place =10

But third place can't be filled using 0..hence we have only 9 options left… (1 to 9)
Ways to fill 3rd place=9
Hence total ways to fill 8 at unit place=10×9
=90