If you look at the letter of the law, then an ATZ doesn't make you exempt from the rules and regulations of the CAS. You're in CAS, so you need permission from the CAS controlling agency. Simple. For inbound traffic that means you need to contact them 10 miles/5 minutes out, for outbound traffic it means you need a departure clearance from them before taking off. (And that is typically arranged by telephone, either by yourself or by the A/G operator, and includes a clearance void time.)
However, this clearly becomes unpractical both the main and the A/G airfield are busy. So what typically happens is that a local agreement between the main airfield and A/G airfield is reached whereby you can enter and exit CAS via a certain route, possibly squawking a certain code, to arrive at or depart from the A/G airfield. And if you adhere to those procedures, you don't have to talk to the CAS operator.
In certain cases, this local agreement is actually enforced by introducing a Special Rules Zone (SRZ) or something similar to that. The example that comes to mind is EHBD which wholly lies in the EBBL CTR. The CTR (class C) extends from the surface to 3000', but there's an ATZ with additional rules, from the surface to 1200' where you are exempt from the normal CAS regulations. But only during UDP for instance - at night it's just one big class C CTR.
These agreements tend not to be generic, but specific to the local situation. They should therefore be listed in the AIP entry, flight guide entry or website of the airfield.
So without knowing what airfield you are talking about, it's hard to give you a specific answer to your question.