For example, rising embraced by primitive with enthusiasm (7)
I believe the answer is:
eagerly
'with enthusiasm' is the definition.
(I've seen this before)
'for example rising embraced by primitive' is the wordplay.
'for example' becomes 'eg' ('e.g.' can be a synonym of 'for example').
'rising' shows that the letters should be reversed in order (going up in a down clue).
'embraced by' indicates putting letters inside.
'primitive' becomes 'early' ('early' can be similar in meaning to 'primitive').
'eg' backwards is 'ge'.
'ge' going within 'early' is 'EAGERLY'.
(Other definitions for eagerly that I've seen before include "With keen anticipation" , "With enthusiasm" , "anxiously" , "Keenly" , "Fervently" .)