Ghostly atmosphere finally invading republic to the west (5)
I believe the answer is:
eerie
'ghostly' is the definition.
(eerie means odd or strange)
'atmosphere finally invading republic to the west' is the wordplay.
'finally' says to take the final letters.
'invading' indicates putting letters inside.
'republic' becomes 'eire' (Irish name for Ireland).
'to the west' says the letters should be written in reverse (letters written right to left or 'east to west').
The final letter of 'atmosphere' is 'e'.
'e' inserted into 'eire' is 'eiree'.
'eiree' back-to-front is 'EERIE'.
(Other definitions for eerie that I've seen before include "Scary, weird" , "Spooky, weird" , "Spooky, unearthly" , "Uncannily strange" , "Sinister" .)