Right
As you already discussed the " immortal " with the staff, i'll add something to it.
Chinese - Shou Lao. Often seen holding a staff and a peach
Japan - Jurojin. Staff and tablet, fan, looks slightly different.
In both cases you might see a stag / deer at his feet. In both cases it is the god of longevity, good health etc
What you have, i think, is Jurojin. Sometimes depicted with a figure on his head
This is a Chinese one of Shou i catalogued a few years ago and although damaged, it still made £1200