The way I found the solution to this puzzle was looking at the numbers and the clues at the same time. Here is the table:
18 86
120 42
46 64
80 8
22 83
The first clue was "The number has two digits". So that ruled out 8 and 120. This is how the table looks now:
18 86
42
46 64
80
22 83
Then I looked at the next clue which said "Both of the digits are even". That meant that 18 and 83 could no longer be there because 1 in 18 and 3 in 83 are both odd numbers.
86
42
46 64
80
22
After that, I read the next clue which was like this "The digit in the tens place is greater that the digit in the ones place." Because of that, 22 is out as 2 has the same value as 2.
The next clue I looked at was the one that said "The ones digit is not in the three times table". That meant that 42 and 64 are ruled out as 2 and 4 are not multiples of 3. This is how the table looks with 2 clues left:
86
46
80
The second last clue read "The tens digit is not double the ones digit". All three of the of those numbers fit the question so nothing changed.
The final hint was "The sum of the two digits is a multiple of five". 8+6=14 so 86 can't be the number and 8+0=8 so 80 can't be the number. That leaves us with 46 (4+6=10) which I think is the Mystery Number.
Thank you for reading!