So I'm looking for further inspiration for an Alesha, Who Smiles at Death commander deck. I was wondering if Boast abilities could be activated when the creature comes in tapped and attacking. I know that it skips over triggers that would only trigger when a creature is declared as an attacker. However, Boast looks different to me since it looks like an activated ability that works as long as the creature attacked that turn.
No. For the purposes of the activation instruction on boast abilities ("Activate this ability only if this creature attacked this turn and only once each turn" [C.R. 702.141a]), a creature that enters the battlefield attacking hasn't "attacked" this way, any more than it has "attacked" this way for the purposes of triggered abilities (see also C.R. 508.4).