Hi Andy,
Yes, this would be a typical process but can be doable with some approach. You can following the steps as below:
1) You will need to setup all channels, channel fields, categories, file upload preferences of site1 within site2. For this process, you might understand the both sites contents management to get better idea.
2) At site1, you will need to write some SQL which can results all the channel entries data and this can be exported in the form CSV or XML. This SQL would result entries field data as well as categories. The exported CSV or XML can be imported into site2 with help of some third party add-ons like DataGrab and Importer. You can do it from PHPMYADMIN.
3) You may be using some third party field types in site1 also which need to care about if can be imported easily in site2 or if needed do some stuff by moving database table of that field type as some field types like Matrix stores data into their own created database table.
I hope, above would give you some idea.