You can complete this mission during chapters 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, and 26. The SWERY 65' is only open from 19:00 to 22:00, and it's closed during the rain. Before you head to the SWERY darts bar to do this mission, make sure you've got a slot open in your weapons inventory.
You'll need to buy the Dart Gun from Richard before heading over the to darts machine. During the game itself, there's one big rule to keep in mind: No matter what, hold your breath and hit the triples on the highest number in each flashing part. The flashing sections pop up on every 3rd dart. If you do this, it barely matters what else you get. If you hit every triple and the final bullseye for the flashing sections, you'll have well over the 2018 needed for the high score.
Just a general tip, save your breath for when you need to hit those flashing triples. Again, as long as you hit those, you won't need to hit much anything else. Of course, hitting a few high singles or better on the normal turns doesn't hurt, so still try for those, but don't waste breath on them.
If you're really having trouble getting the high score on the darts, use a stabilizer before the round.