how long does take to gan access to game pass quests after join game pass on the microsoft...

It shouldn't take too long but if you have recently signed up it could possibly take a couple days to process. If you still have issues after a few days then try and contact support here.

Do you mean how long do you have access to the game with Game Pass? You can play it as long as you have a Game Pass subscription and as long as the game is included in Game Pass.

Cant send a message to the link you provided.. But this is whats been going on I can’t access game pass achievements on my Xbox one and the game pass app does not show the quests

The Metro series quests are NOT apart of the Game Pass quests. Metro Exodus is not included in Game Pass, so that quest is not applicable when it comes to completing "Game Pass quest master" quest. I verified this through offical support, and checking
that it does not show up in the applicable quests in the Game Pass app on iOS or Android.

The Metro Exodus quest is just a seperate bonus promoting Metro Exodus, 2033, and Last Light. You do not need to complete it to finish the Game Pass quests.

I just verified via the Game Pass app on Android. It does NOT include the Metro series quests via that app, so will not be required to complete Game Pass quest master quest.

Sometimes they add promotional reward quests, and the Rewards dashboard lumps all the quests together, and doesn't seperate the Game Pass specifics. You shouldn't need to purchase that game to complete the Game Pass quests. You can check your progress
with the Game Pass app available on iOS or Android.