In theory, Reverse Address Resolution Protocol (RARP), defined by RFC 903, can be used to map a device MAC address to an IP address. In practice, RARP requires both a RARP daemon running on the AP and a RARP client to query the daemon. RARP fell into disuse many years ago, so you are unlikely to have either of these at your disposal.
However, if you have access to a nearby switch or router, plug the AP in and check the ARP table on the switch/router (e.g., arp -a). If the AP is sending any traffic at all, there will probably be entries in nearby ARP tables. You can often cause a device to generate a little traffic on the Ethernet by rebooting it -- for example, BOOTP or DHCP renew requests or proprietary inter-AP broadcast announcements. If that doesn't do the trick, you can try ping-scanning the entire subnet if you know the subnet address but not the host IP. If you don't know the subnet, try RFC 1918-defined private subnets. If you have access to the Ethernet but no nearby devices, use a LAN analyzer like Ethereal to watch for traffic from that MAC address.