Information here comes from the wiki and my personal play experience.
Each Act has 3 potential bosses. The Bronze Automaton is one of the possible bosses for Act 2. You can see if you're going to fight the Bronze Automaton from the map. If there is a core with two gear/cog things around it at the top of the map, that means you will fight the Bronze Automaton and can prepare accordingly. The good news is, this boss is pretty easy if you know what you're doing. The bad news is, RNG can work very hard against you in this fight.
First, we should talk about getting to this boss in the first place. If you're going specifically for this achievement, you can keep restarting until you see his symbol at the top of the Act 2 map. However, you will definitely face him many times while playing the game naturally. Most of the time, you'll want to prepare by improving your deck. In Act 2, you'll do this by choosing beneficial paths with fights for cards or shops for cards and relics (being careful to avoid losing too much HP). However, you can safely lose some HP since there is always a rest point before the boss fight. In Act 2, your deck should be coming together to form a particular strategy or achetype. So, try choose cards and artifacts that fit this archetype, without being afraid to pass up cards that don't work with your strategy. Do not fall into the trap of always taking a card, because this will only bloat your deck and make it worse.
Once you are face to face with the Bronze Automaton, you will see he has 300 HP. He attacks in a pattern, so I am going to list it below, along with the strategy at each step.
1. Spawn Orbs - On the first turn of the fight, the boss will summon two bronze orbs. Use this turn to apply buffs and debuffs and attack. We will want to kill both orbs as quickly as possible, so prepare accordingly. The orbs will most likely steal the two rarest cards in your deck next turn, so now is the time to use them if appropriate. Also note that the boss begins with 3 artifact, meaning he will nullify the first 3 debuffs you give him (so prioritize weaker ones first).
2. Flail - This attack deals 7 times 2 damage, so gain 14 block if you can without wasting your entire turn on it. The Bronze Orbs will have between 52 and 58 HP each. The boss will not bring them back, so you should focus on killing them first. There is a 75% chance their first move will be Stasis, which steals the rarest card in the draw pile, or from the discard pile if the draw pile is empty. Take note of which cards are likely to be stolen and plan to kill the orb with card you need most first.
3. Boost - The boss gains 3 Strength and 9 Block. Here's where the real trouble begins. Think of these strength boosts as your time limit. The boss will overwhelm you with strength if you take too long. I suggest you do your very best to have one of the Bronze Orbs dead by the end of this turn. When they die, you get the card that they stole back in your hand, so you can use this knowledge to your advantage by saving energy for that card. Once they have your card, they'll start doing attacks for 8 damage or giving block to the boss. Either way, kill them ASAP.
4. Flail - With the newly added Strength, deals 10 damage 2 times. If you can, use 1 card that gives you block here to mitigate the damage and spend the rest of your energy on killing the last Bronze Orb. If you haven't killed either of them, it may be better to just tank the damage to finish at least one of them off. Don't waste any time overblocking unless you're already dangerously low on HP.
5. Boost - Same as turn 3. Focus fire orbs if they aren't dead. Otherwise, it is time to focus your buffs, debuffs, and damage on the boss. With his minions dead, he has a lot less armor to protect him, but his damage output is still very high. Luckily, you have a little more freedom to use block cards to mitigate damage with them gone. But don't forget that his damage will only increase, so killing him fast is top priority.
6. HYPER BEAM - That's right, he knows a Pokemon move. After 2 boosts, it is going to deal a whopping 51 damage. Chances are that at this point, you can't survive the full brunt of this attack. Apply weakness if you can (but don't waste the energy if he still has any of his artifact buff) and use the rest of your energy on block. Ideally, he is now near half HP or less.
7. Stunned! - Just like on Pokemon, the boss has to spend a turn recharging after that devastating attack. He's more vulnerable than ever on this turn, so really lay it on him. Remember that you are in a race against time. At this point, you likely can't survive many more attacks, and almost certainly can't survive another hyper beam. Give him every of bit damage you've got.
From here on out, the Bronze Automaton will repeat steps 2-7, but with his strength buff (which will increase further). Mitigate the damage as best you can without slowing your damage too much. Once you're at this stage, you're probably getting very close to death and you need to focus on damage, damage, damage. By the time he uses Hyper Beam again, there is almost no hope whatsoever of survival.
So, to reiterate: with most decks, this fight is a race against time (like most fights in Slay the Spire). After the minions are summoned, you need to kill them as quickly as possible because they will steal your cards, but they do not respawn. You need to kill the boss before he uses Boost for a third time or as quickly as possible after. You probably won't last long past that point.
One final tip: save your best potions and cards that debuff for AFTER you have gotten rid of the 3 artifact if you can. For example, if you have the weakness potion, you should get rid of the boss' artifact with cards before he uses HYPER BEAM, then use the potion on the turn that he intends to use it. If you do this, you will also weaken his next Flail.