I think you are:
I’m an island domain, and where strong drink is found,
There is something called Nirvana Island: The Last 47 Days. There is also a Nirvana cocktail.
Yet I’m baffled by cricket, and silent (no sound).
There's a Nirvana Cricket Club. The band Nirvana disbanded after Kurt Cobain died.
I’m a person of faith, and devoid of all strife;
In some Buddhist traditions, nirvana is the achievement of stillness/nothingness.
I am "the Way”, and one’s stance after life.
The Middle Way is a description of the Noble Eightfold Path, which you follow to achieve nirvana in the afterlife.
I’m a small, weighty rock, and unchanged from the past.
Nirvana is the name of a rock band. The band dissolved, so its music remains unchanged.
I am not going anywhere fast!
Nirvana is a sort of transcendent state where you are infinitely freed from the pushes/pulls of everyday life. Therefore, once you achieve nirvana, you won't be going anywhere.