You can activate Grave-Shell Scarab's first ability, as with most other activated abilities, any time you have priority, which normally means during any player's turn, even an opponent's (C.R. 116.1b, 116.3a, 116.3d). That's because the ability has no restrictions on when it can be activated (within the meaning of C.R. 602.1b), and it isn't a loyalty ability as it has no loyalty symbol in its cost (C.R. 606.2, 606.3).
EDIT (Apr. 25): Deleted a portion of the first sentence for correctness reasons.
The previous replies have completely addressed your question in technical terms. I would like to add for your own education, that you appear to be confusing timing restrictions with summoning sickness restrictions.
Unless otherwise stated explicitly by the ability itself (for a keyword ability such language may be in the rules rather than printed on the card, e.g. equip), all activated abilities have what is called "instant timing". You may activate them anytime you could cast an instant.
Summoning sickness restrictions are not timing restrictions, they are action restrictions.
A summoning sick creature cannot be declared an attacker nor can you pay for any tap/untap symbol costs associated with that creature's own abilities.
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(1), Sacrifice Grave-Shell Scarab: Draw a card.
Since this ability doesn't require the creature to tap, does that mean I can activate it during my opponent's turn?
EDIT (Apr. 25): Deleted a portion of the first sentence for correctness reasons.
Unless otherwise stated explicitly by the ability itself (for a keyword ability such language may be in the rules rather than printed on the card, e.g. equip), all activated abilities have what is called "instant timing". You may activate them anytime you could cast an instant.
Summoning sickness restrictions are not timing restrictions, they are action restrictions.
A summoning sick creature cannot be declared an attacker nor can you pay for any tap/untap symbol costs associated with that creature's own abilities.