Is this out of date? My iMac was made in 2012

In principle, every AirPrint printer model should be compatible with macOS 10.11 El Capitan, if the printer and the computer are connected to the same network (Wi-Fi router). The AirPrint technology as such may work with operating systems as low as Mac OS X 10.7 Lion. However, if you study the specifications for a certain AirPrint model, you may notice higher system requirements. One reason for this could be that the printer manufacturer provides drivers for USB connections, and/or special software, for example, for setup, ink management, and similar purposes.

An HP DeskJet 4155e All-In-One Printer is listed in About AirPrint - Apple Support. An AirPrint model can be used without the installation of additional drivers, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router). In this case, you could perhaps at least carry out a test with AirPrint, using a manual setup. For example, if supported by the router, WPS (Wi-Fi Protected Setup) could be tried for the Wi-Fi connection. "Free" ink subscriptions, et cetera, may not be possible, though. You would have to check the details in any agreement in connection with the purchase, if applicable.

If necessary, it may be possible to upgrade the operating system on this iMac. Sometimes, macOS 10.14 Mojave or higher is required for setup apps from a printer manufacturer.

Thank you for the information.

Is that Wi-Fi from the tower, or possibly some kind of mobile/cellular service? How do you receive it? Directly by your computer, or Is there a special receiver in your house? If the latter, any information about the receiver type/model and its available ports?

Generally speaking, it is not difficult to create a small, local Wi-Fi network for printing purposes, thus allowing AirPrint to be used. One way is to simply use an old Wi-Fi router (you can probably find a second-hand router in a thrift shop for a few dollars). Almost any model should work. The section of interest is the WLAN/LAN. The WAN (Internet) port can be left unconnected.

Hello VikingOSX,

Yes, that is true. See also https://www.office.xerox.com/latest/XOGWP-18U.PDF.

However, in this case, the 4155e printer seems to be (USB and) Wi-Fi only.

https://www8.hp.com/h20195/v2/GetDocument.aspx?docname=c08289618

The iMac could of course be wired by Ethernet to one of the LAN ports of a router, as an alternative to a wireless connection (Ethernet is often a good solution).

The main problem (?) here is the printer setup limitation (macOS 10.14 Mojave/HP Smart). The "+ service" with "free" ink and an always active (!) Internet requirement does not appear to be possible. The OP will have to check any agreements.

I guess that a manual setup would work, though (WPS or via the embedded web server). AirPrint should be possible via a separate Wi-Fi router, as described above.

Hello again,

It should at least be possible to use an HP Envy 7640 as a copy machine (directly from the printer's control panel, without a computer connection).

if, despite the difficulties so far, you are prepared to give one of the printers another try, do not hesitate to post back.

Addendum

>macOS 10.11 El Capitan

Should be OS X 10.11 El Capitan

How to get old versions of macOS - Apple Support

Add a printer on Mac – Apple Support (UK)

AirPrint supports Wi-Fi and wired Ethernet devices. Working with the latter connection here for my laser printer.