I think it really breaks down to your traffic. The slower the hardware, the slower (the already slow) page generation. It’s not specific to WP, it’s the same for all large php scripts.
If you get a server hit every now and then, as on a dev box or a family blog, it’s no big deal. It’ll just spit out pages more slowly. If you get concurrent hits on a regular basis, your site will get knocked off of the web quite easily.
Also note that many a low end shared host stuffs multiple sites on low end hardware. It’s obviously not as low end as what you linked to, but I wouldn’t expect that much of a difference in performance as compared to an overstuffed shared server.