So it depends on the details. You can push (Publishing) objects/fields to a connection. The connection (your partner) then needs to receive (subscribe) to create the case. There is setup on both ends.
The challenge with cases is the lookup fields. If you're not sharing the same Accounts and Contacts with Salesforce to Salesforce you cannot fill in the look up fields. The ID of the account is not the same account in your partner's database. To compensate you can either share all Account/cases -- or you can create formula fields that will display the name of the account/contact and any other available information. Your partner will then have to use that information to determine the correct records for their look up fields.
Once you have the connection setup you mark you want to sync a particular case. After that, it's up to your partners to accept the record and have the right fields subscribed and mapped