Rising expectation about century being a significant period (5)
I believe the answer is:
epoch
'a significant period' is the definition.
(I've seen this before)
'rising expectation about century' is the wordplay.
'rising' is a reversal indicator (going up in a down clue).
'expectation' becomes 'hope' (hope is a kind of expectation).
'about' is an insertion indicator.
'century' becomes 'c' (common abbreviation).
'hope' going around 'c' is 'hcope'.
'hcope' in reverse letter order is 'EPOCH'.
'being' is the link.
(Other definitions for epoch that I've seen before include "Historical age" , "Division of a geological period" , "Long period of time" , "Period of time marked by a great event" , "remarkable time" .)