OK, well, according to the HP website, there are no downloadable drivers for that product, and that basic drivers are already included in win7.
So it should 'just work' when you plug it in, but with a reduced level of functionality: the various one-touchbuttons won't work, for example.
Whether Windows Update carries better drivers that were released in-box with win7, I can't say.
You could try moving the printer to an on-line box and seeing whether the updated drivers are substantially different to the existing in-box ones.
I've had mixed results with the various driver download sites.
Mostly back in the dial-up days, trying to get the connexiant chip-du-jour do work with the driver-variant-du-jour.
I've never had anything mallicious, but a lot of crud to wade through.
HP support for 'consumer' level stuff ( like this inkjet ) and 'business' level stuff ( like the laserjets ) is rather different.
With consumer-level stuff, they put very little effort into supporting it in newer versions of Windows. They'd rather you just chucked it and bought a new one. They realise that won't wash with corporates with an existing fleet of laserjets, so they carry OS support for these products much better.
I'm going to say that you're probably stuck with the in-box driver functionality.