Choosing my first oscilloscope
Yes, the Owon should do okay for all the mentioned computers. 20MHz bandwidth is a perfectly good choice for your first scope, and will cover most of the "less complex" and older digital electronic components, for example small microcontrollers (e.g. PIC10, 12, 16, and 18F and similar Atmel, TI, etc offerings)  

The highest clock speed out of the computers mentioned would be probably the Ataris - you don't specify which one but for example the Atari ST used a motorola running at 8MHz, and a later model used a 16MHz processor.
The C64 and Apple IIe used ~1MHz clock speed, so obviously they are no problem.  

Note that most of the signals you will be looking at will be much slower than the clock speed, so even if the clock speed is above your scope bandwidth it doesn't necessarily mean you cannot use it. It just gives a very rough guide, as you know (almost) for sure all the signals will be slower than the main clock speed (barring things like wireless peripherals or video ICs that may generate their own high speed clocks) 

Another thing thing to note is although the bandwidth is given as 20MHz, the sample rate is only 100Msps (mega samples per second), so a 20MHz square wave will not look very square at all (as you will only get 5 samples to recreate one cycle of the waveform).
Usually decent scopes are specified with a bandwidth of around a 10th of the sample rate, so over 10MHz this Owon will not be ideal. Looks like they were pushing the specs a bit to make it sound a better.
However, they are pretty good scopes for the price - I have a 200MHz later SDS model which samples at 2GHz, so it looks like they may have reconsidered about the sample rate vs bandwidth specs.  

Out of interest, how much will you be paying (just to make sure it's a reasonable price)

EDIT - $150 (I assume USD) sounds pretty reasonable for a new DSO of this spec. Here are a few interesting alternatives:

TEKTRONIX 2235A 100MHZ OSCILLOSCOPE - $175 (analogue scope) USED - you can get plenty of higher bandwidth analogue scopes for the same price. The downside with an analogue scope is you cannot save the waveform, or see before the trigger (pre-trigger) or do FFTs/waveform arithmetic. Still very usable though - even though I have a good DSO I still use my analogue scopes regularly.  

Tektronix TDS 1002 Two Channel Digital Storage Oscilloscope 60 MHz 1 GS/s LCD Bid currently at $172 (USED) - probably go for double this but definitely worth watching, a nice scope.

HP 54542A Oscilloscope 500MHz / 2GS/s , 4-Channel - Just for interest, be nice if it went cheap...

Siglent SDS1062C Digital Oscilloscope 1Gsps/60MHZ DS1052E - £189 (GBP) NEW - There are quite a few scopes around this price up to 1Gsps/100MHz or so, which is four times the bandwidth of the Owon PDS5022S. In case you want to spend a bit more.