Negative.
Well, it would stop Fatestitcher, but not in the way you are asking. Flashback and Unearth are not activated abilites, so needle has no effect on them. Needling fatestitcher will however keep him from using his Tap/Untap ability when he gets into play.
502.84 - Unearth
* 502.84a - Unearth is an activated ability that functions while the card is in a graveyard. "Unearth [cost]" means "[Cost]: Return this card from your graveyard to play. It gains haste. Remove it from the game at end of turn. If it would leave play, remove it from the game instead of putting it anywhere else. Play this ability only any time you could play a sorcery." [CompRules 2008/10/01]
Unearth apparently is an activated ability (just found out too), so Needle would stop that.
Look for the colon :
If there is no colon, it is not an activated ability.
Fatestitcher
Creature - Zombie Wizard 1/2, 3U (4)
: You may tap or untap another target permanent.
Unearth
(
: Return this card from your graveyard to play. It gains haste. Remove it from the game at end of turn or if it would leave play. Unearth only as a sorcery.)
Ancient Grudge
Instant, 1R (2)
Destroy target artifact.
Flashback
(You may play this card from your graveyard for its flashback cost. Then remove it from the game.)
