What is the correct, usual way in English to say "have a shot (of a drink) in praise of your youth days"?
“To the riotous enjoyment of X.” How to Give a Toast
As in your example:
have a shot (of a drink) in praise of your youth days
I would suggest:
“To the riotous enjoyment of your youth! Or "Let's drink with riotous enjoyment to your youth!" (with glasses raised ... toast ... drink ... then all glasses thrown aside!)
Seriously, one does not have to throw the glass or container!"
Good article on all thing 'toast'! There many examples of toast for all occasions! Just stand up, raise you glass and speak you mind ... carefully.