Rising reputation's right concerning Viking (8)
I believe the answer is:
norseman
'viking' is the definition.
(I've seen this before)
'rising reputation's right concerning' is the wordplay.
'rising' is a reversal indicator (going up in a down clue).
'reputation's' becomes 'names' (name is a kind of reputation).
'right' becomes 'r' (common abbreviation).
'concerning' becomes 'on'.
'names'+'r'+'on'='namesron'
'namesron' in reverse letter order is 'NORSEMAN'.
(Other definitions for norseman that I've seen before include "Scandinavian" , "one attacking north-east?" , "invader" , "A viking, perhaps" .)