Is it at all possible to use an older (2013) imac as a display with a new (2021) macbook pro?

No sorry, none of the latest Mac’s support using a qualifying iMac as a display in Target Display Mode.

carefully read > Use your iMac as a display with target display mode - Apple Support

• The other Mac that you're connecting the iMac to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.

+1 no but you can try a third party tool such as Luna display. Personally if it were me I buy an external display. VERY simple to setup, doesn't require expensive cabling or software and its dead reliable.