What are the obligations of a business partneras the signer on a contract?

It depends upon the terms of the contract and also what the contract was for; also, how the transaction was done. 

First of all, when you bought the business, did you buy the business structure (e.g. the shares of a corporation) or did you buy the assets? If you bought the structure and if the contract was not personally between #1 and #2, but rather between #1 and #2's business, which you bought, then #1 would still be obligated, since he is obligated to a still-existing entity, which you now happen to own.

However, if either you bought the assets of a business, but not an existing business structure; and/or the contract was not with a  business structure in the first place, but rather between two individuals, then #1 would not have to follow the terms of the contract with you unless the contract was validly assigned--transferred--to you.

As to assignment: many contracts are by their terms are readily assignable, and all that's required is execution of the correct paperwork. Others specifically state they can't be assigned; if that's the case, then #2 cannot transfer it to you and you'd need to negotiate a new contract with #1.

Also, certain obligations under law are not generally assignable. These are typically personal services obligations. For example, say that #1 had contracted with #2 that #1 would do his accounting, be his personal trainer, etc. I do not believe that you could step into #1's shoes and take over for those personal services. (If the contract had been with a business to provide, say, accounting or training services, that's different; we're talking here about a contract for personal services between two people, when #2 directly contracted with #1 to provide the services.)