I had an 81 in 35 Rem a few years back . It had a weaver side mount on it , which is about your only option . The 81 has been out of production for many years now , so there aren't any mounts offered by the major manufacturers . I think Brownell's carry the weaver mount , but I may be mistaken .
If your 81 is in good shape , I'd probably leave it original . They do have some collector value now , and any holes in the receiver will lower the value . I'm not saying don't do it , but there are a lot of easier options out there . What round is your 81 chambered in ?