You can do this forward with your ASA, or you could dual-home the new host for a while (this would be my choice - it's the easiest solution), or you could rig up a proxy like Bart De Vos suggested.
None of these this solutions will solve your underlying problem though: Presumably you need this customer to get off that IP for some reason, and by keeping it working you provide them no incentive to change their DNS.
I know it's not what you want to hear, but you need to give your customer a hard deadline by which time thy MUST change the A record, because the IP will go away at
hard deadline + X hours. (If you have the time you should ideally send this notice to them in writing, certified mail, return receipt, yadda yadda yadda, so there's no disputing that they were notified). If they fail to comply, let their site break.
I've been through 3 IP migrations from the ISP side, and 2 from the client side. In my experience a customer who does not respond to repeated polite requests to change their addresses will NEVER do so until their hand is forced. (And to be fair, many ISPs will hang on to address blocks their delinquent customers are still occupying, right up until the point their RIR says "You can't have any more IPs until you stop advertising those! They're not yours anymore!")