Video decoding is dependent on hardware acceleration, CPU speed, and software. RAM and hard drive/SSD
don't matter much. Network speed only matters if streaming without allowing the video to buffer.
As for hardware acceleration of video decoding, the GMA3150 graphics in the Mini 1012 has essentially no ability. So, if you don't have the optional Broadcom decoder, video decoding falls to the CPU.
The Atom N450 CPU is slow compared to other modern CPUs, but it still has enough power to decode 480p h.264 videos without problem (h.264/AVC is a popular video encoding). But the Adobe Flash plugin used by YouTube and Hulu and most web video sites is poorly optimized, so even 480p video can run poorly through it.
If you can download the Flash videos and play them in a separate player, they should work perfectly. That's what I usually do (on Linux at least, many Flash videos get automatically saved to temporary files, I just open those files in a separate program).
As for 720p h.264, it's very demanding. You can look into CoreAVC, but it won't help Flash web video playback in the browser, just Windows Media Player and similar programs.
If you want to play 720p h.264 in VLC or Mplayer, or in a normal Windows player via the ffdshow decoder package, look into disabling the "loop filter". It sacrifices quality, but improves speed.