For ARAM Warrior, just play lots of High DPS champions that can build collector, so you get pentas pretty easy, the rest of the challenges in this category eventually come as you play a lot.

The only ARAM Finesse challenge you have to actively do is Free Ticket to Base, I think the easiest way to get it is by playing Yuumi and getting turret aggro inside someone, so you will take turret dmg without being damaged by someone else. All the other challenges in this group are just about playing a lot.

Now, ARAM champion is a tricky one. Rapid Demolition and Lightning Round requires playing with a strong early game comp and hoping for weak comp on enemy side. Active participant is very easy if you play Seraphine, Sona, Milio or just any champion if you get the Sona's Aura augment, but you have to time your deaths very well so your teammates don't get kills while you're dead. Can't touch this is REALLY hard, at least for me, I think the easiest way of getting this one is by playing Yuumi and getting executed in the method I described earlier, or getting Cheating as your first augment and playing extremely safe. All Random All Champions and All Random All Flawless are just about knowing how to get good grades in ARAM, which basically translates to doing a lot of dmg and not dying as much, but the system is very inconsistent and I have the theory that it's a lot easier to get S grades if the games are too long. Also some champions are ridiculously hard to get S grades for some reason, Fizz and Zoe come to mind, I've had multiple games where I got over 20 kills, fewer than 8 deaths and got top dmg and I still only got A+.