In order to get this achievement you have to do 2 things:
On day3 when miss Jenkins asks you to take the cat to the vet choose the second option: " Sure I’ll get right on it! Here, little kitty come on! " Then go to the bait shop and talk to Mr. Mackey to deliver Mortimer the cat.
On day4 go to the bait shop again to grab Mortimer the cat and go back to miss Jenkins. After the talk the achievement will pop
For a Fast Achievement Guide check the link below:
Timestamps:
00:00 Intro + Day1
01:39 Day2 Intro
01:55 Day2: Higher Ground Achievement
02:22 Day2: Deliver Mail
04:39 Day3: Deliver Mail
05:24 Day3: Cat Lover Part 1 of 2
06:26 Day3: Deliver Mail
07:01 Day3: Workoholic Part 1 pf 4
07:19 Day4: Tower Watcher Achievement
07:57 Day4: Cat Lover Achievement
09:12 Day4: Movie Carrier Part 1 of 5
09:44 Day4: Deliver Mail
11:20 Day4: Movie Carrier Part 2 of 5
11:35 Day4: Workaholic Part2 of 4
11:46 Day5: Shutter Bug Part 1 of 3
12:19 Day5: Movie Carrier Part 3 of 5
12:51 Day5: Metal Detector Achievement
13:59 Day5: Deliver Mail
15:21 Day6: Shutter Bug Part 2 of 3
15:42 Day6: Movie Carrier Part 4 of 5
16:51 Day6: Deliver Mail
19:09 Day7: Workaholic Part 3 of 4
19:35 Day8: Interrogation
19:56 Day8: Shutter Bug Achievement
20:51 Day8: Movie Carrier Achievement
23:31 Day8: Deliver Mail
25:01 Day8: Free Spirit Part 1 of 5
25:46 Day9: Deliver Mail
29:03 Day9: workaholic Achievement
29:48 Day10: Deliver Mail
32:11 Day10: Free Spirit Part 2 of 5
32:25 Day11: Deliver Mail
33:54 Day11: Free Spirit Part 3 of 5
35:20 Day12: Deliver Mail
37:40 Day13: Deliver Mail
40:35 Day14 – Repeated 3 times
40:57 Day14: Free Spirit 4 of 5
41:10 Free Spirit Achievement (Ending 1)
42:31 Life in a Fast Lane Achievement (Ending 2)
43:29 Hometowner Achievement (Ending 3)