The prestige class arachnomancer (Drow of the Underdark 65-8) says
You gain access to the Spider domain and the granted power associated with the domain (rebuke spiders as an evil cleric rebukes undead, a number of times per day equal to 3 + your Cha modifier). (65)
Hence a level 1 arachnomancer rebukes spiders like a level 1 cleric rebukes undead—no matter what other abilities the arachnomancer possesses—, and that's terrible.
The prestige class earthshaker (Dragon #314 57-9) says that the creature
gains the Earth domain's granted power, substituting his earthshaker levels for cleric levels. (58)
And, yes, that's also terrible, but it's worse because while expected entry into the prestige class arachnomancer is at about character level 5, expected entry into the prestige class earthshaker is at about character level 9. Your granted power isn't just bringing almost nothing to the table, it's actually insulting.
Extra domains in general
Although earthshaker predates Complete Divine, that version of the arachnomancer was released long after Complete Divine.1 Both, however, seem to follow similarly the Complete Divine's vague rules on Extra Domains which, in part, says
If a noncleric enters a prestige class that allows access to a domain, the character still gains access to the domain. She can use the granted power bestowed by the domain normally. (20)
Complete Divine, of course, doesn't actually say what normally means here, but, apparently, based on the anecdotal evidence from the above two classes, the game takes normally to mean here that the creature treats its levels in the prestige class as cleric levels, combining with nothing unless the text indicates otherwise. Which, let me say again, is terrible.
Negotiate with the DM
I suggest asking the DM to allow the arachnomancer to rebuke spiders like an evil cleric of twice the creature's arachnomancer level. While that's still terrible, it does sort incentivize staying in arachnomancer, because it (ahem) bugs some DMs when players have a bunch of a different classes on their character sheets. Creatures likely enter earthshaker too late for this quick-and-dirty fix to do any good, and thrice (maximum 20th) earthshaker level might be overreaching, but it couldn't hurt to ask.
1 The original arachnomancer was, confusingly enough, introduced in Underdark, an early but still 3.5 product.