It is actually very simple to upgrade the store. You only need to click on the part-time job page first, and you will see an upgrade button in the upper left corner. And if you click in from the task side, this button will even be marked with the task.
After finding this button and clicking it, you will be able to see the materials and gold coins required to upgrade the store. If your mission requires you to upgrade, then the materials required for upgrading at this stage are all available to you. If the material is not available, it requires completing other tasks to obtain the material formula.