Can I use ADB (USB Debugging) with cable that has USB-C connector on phone side and USB 2.0 Type A plug on the computer side? It should be able to transfer data.
The cable you describe is a standard USB converter cable (USB-A to C).
A lot of manufacturer ship such a cable together with a new phone. You can use it to establish a USB 2.0 (and may be 3.0 if the device supports this) connection between your computer and your phone.
Like any USB cable it is agnostic to the data it transmits, therefore ADB and MTP (file access) will work as expected as long as the used computer is correctly configured.