The 8051 don´t implements a FIFO stack.
If you want one yo must do it ins soft or hard.
In your problem i think that a soft one is enough.

How to do:
Make two pointers at external mem.
If you have a derivate with two DPTR you can use it.
Make two routines one 'pop' and one 'push'

pseudo code for push:
movx @pointer_push,a
inc pointer_push
If Is > than the top of stack
pointer_push=start_stack
If is = pointer_pop
set flag 'busy'
ret

pseudo code of pop
movx a,@pointer_pop
push a
inc pointer_pop
If Is > than the top of stack
pointer_push=start_stack
If is = pointer_push
set flag 'empty'
pop a
ret

Regards.