Does capacitor package size matter?
The first capacitor is a 0402 package, but I can't solder that. Can I replace that with a similarly specified 0805?


First, 0402 isn't that hard to solder. Get a good pair of tweezers and give it a try.

If you do need to substitute, the main advantage of 0402 over 0805 is lower package inductance. It's probably not critical in this application, but in case you see trouble with the 0805 solution, you might try five 1-uF 0805 capacitors in parallel instead of a single 4.7 uF part. 

Another issue is availability. There's a worldwide shortage of multilayer ceramic capacitors (MLCCs) at the moment, and the big vendors like Murata are cutting production of larger sizes to focus on smaller sizes. So you may have trouble finding 1 uF in an 0805 package, or pay more for it than you expected to.


  The second capacitor is a 1206 package. Does that package size matter? Would a 0805 suffice (just out of curiosity)?


Going down in size, the main risk is probably ripple current rating. If you can't find an 0805 with the same ripple current rating as the 1206 you're replacing, again you might need to parallel multiple smaller values (but then this could affect the regulator's stability, if they're counting on the ESR or ESL of the capacitor package to provide a pole or zero in the loop response at a certain frequency, so you might have to experiment)

This doesn't mean the substitutions totally won't work. In either case there's a good chance you'll have no problem. But you'll want to know what could go wrong and check for it when you build your circuit.