Q: How much Swingarm up-and-down play is acceptable?
A: The short answer is little to none! You don't want any up-and-down or side-to-side play on the swingarm. However, it's also understood that the swingarm is a bolted on part thus overtime loses strength. If you get a chance check out the difference between a brand new bike and one that's seen its share of Motocross tracks or the woods.
You'll notice a much stiffer feel to the new bike and probably some perceptible play on the old bike. Perceptible is the key word. As the bike ages and the bolts weaken you'll feel some very subtle play but if it's significant you'll need to address it by greasing the swingarm and linkage bearings. Tightening the bolts won't have any effect on the play. If the greasing doesn't work then you'll need to replace the bearings. For more information see How To Replace Swingarm Bearings.