Here are two ways for a call from a previously unknown number to reach the AA, while still having normal calls ring your home phone.
One method is to get another DID, setting the OBi to route all its calls to the AA. In many countries, free DIDs are available, or you may be able to leverage existing SIPBroker, iNum, Localphone or Rebtel access numbers.
The other way is to call through a gateway that sends a caller ID known in advance, that the OBi could recognize and route to AA. Depending on calling country, GV or Rebtel may be suitable.
With either of the above, it would be possible for a stranger to stumble upon your setup, so you would have to protect the AA with a PIN. Unfortunately, you would need to use the PIN, even when calling from your cell phone, though you could program the cell phone contact entry to send it automatically.
If you provide some additional details, we may find a better way for you to call, possibly not involving the OBi.
What services are presently configured on your OBi?
What countries/cities will you be visiting?
What free calls, if any, are available from the hotel room phone (toll-free, local, national, etc.)?
How do incoming calls to your room work (DID, IVR with DTMF, must speak with operator)?
Does your cell phone work there? Local SIM? Free incoming calls?
Have Internet access? Good enough to use VoIP?
What countries will you be calling? How were you planning to route those destinations on your OBi?