How to pass credentials to a linked server in another domain?
You probably need to set up Kerberos to handle delegation. This requires setting up an SPN for your SQL Server as well. The following question and sole answer give more details:

SQL Bulk Insert Impersonation issue

The main documentation links in that Question and its Answer are:


How to Configure the Server to be Trusted for Delegation
Register a Service Principal Name for Kerberos Connections
SQL Bulk Insert Impersonation issue


Also, I'm not sure if this info will be useful or not, but it might help to have the Login and Password for the account on both domains to be exactly the same. I know that when you do not have Active Directory set up but want to share folders between two systems without having to log in, it works if you use the exact same Login with the exact same Password.