How can I add arc-fault protection to circuits in this panel?
Any electrician who looks at this would consider that subpanel 'maxed out' ... Not necessarily due to the technicalities of the code to squeeze another circuit or two in with complex tandems but because of the amount of wire packed into that small box and the extreme of things taken to extremes - meaning the multitude of tandem breakers. Generally tandems are used for squeezing in 'one more' or properly upgrading a bath or kitchen on a tight panel. While possible to expand this box is a bit fully loaded as it is. 

I would suggest getting a larger subpanel and trying to keep as much as possible as single, standard breakers. You can keep the feed wire and originating circuit breaker (in other panel) the same as long as you're not planning on adding more load. But a new metal panel box is going to be a similar price to a number of $20-30 complex tandems and tandem arc fault breakers.

Here's a subpanel for $75