Are you saying you want the ASA to answer a request as well as forward the request back to another host on the same interface? i.e outside interface? If so, then yes you can.
e.g:
20.20.20.20 = your old DNS
10.10.10.10 = ISP DNS
command:
static (outside,outside) 10.10.10.10 20.20.20.20 netmask 255.255.255.255
you will also need the following command:
same-security-traffic permit intra-interface
this command allows traffic to enter and exit the same ASA interface.
you will also need to configure your access-lists to allow the traffic on the acl applied to the 'outside' interface.
access-list acl_outside permit ip any host 20.20.20.20 eq 53
COMMAND SUMMARY:
static (outside,outside) 10.10.10.10 20.20.20.20 netmask 255.255.255.255
same-security-traffic permit intra-interface
access-list acl_outside permit ip any host 20.20.20.20 eq 53