Spartacus Legends

For this achievement you must win a duel with all 8 fighting styles in the game. As you progress through Legends Mode there are multiple fighting styles you will come across when recruiting different gladiators. Fighting Styles can be attained when you recruit Gladiators that use the said style. Once you defeat Legendary Warriors and other Gladiators, you will eventually be given access to recruit a gladiator that has a specific fighting style, it will be addressed right next to their name. You can recruit so long as you have an open slot, however extra slots will cost you silver. Below is a list of every fighting style:

Where to Fight
Simply win a match using all of the mentioned fighting styles above. The easiest place to win matches is the Extremus District. Go there and play one of the challenges for an easy win with said fighting style. To find Gladiators who use these fighting styles, you will naturally make them available to recruit as you conquer districts in Legends Mode.

Gladiator Slots & Coin
Once you buy extra slots, you can recruit Gladiators with different styles for 420 silver or more. It costs around 3000 for the first slot, but the more slots you buy the more expensive they become. It starts to jump to 6000 silver after a while. Then, once you have 5 total slots it will start costing 17 Gold Coins or more after purchasing more slots. If the style you're looking for is not present, return to recruitment in 30 minutes for a new set of Gladiators to choose from. The recruiting station will refresh every 30 minutes as well as every time you level up.

If you have enough Gold Coin you can also refresh the recruiting list whenever you want for 5 Gold coins each time. If this is out of your means, you may have to be patient with this as it can be luck based on which styles you get. Just check back the recruit lineup in between conquering districts.

You can also make expendable use of your Gladiators for this achievement to save slots and coin. Once you have won a fight with the style you need you can either intentionally lose a fight as them and then refuse to revive them or you can delete them while in menu selection by scrolling over to the far right.

Note: For the Unarmed fighting style, you simply can un-equip a weapon from any Gladiator by highlighting their weapon and hitting the A button. Once you do this they will become the "Unarmed" fighting style, it will say so in the right corner of your screen. Win a match with them like this to count towards the achievement. Another way to un-equip them is to go to their weapon, scroll to the very left section, and hit A.

This is actually easier, and cheaper than you might think. Earn enough coin to unlock a few extra slots, making sure that every new recruit you go for has a new weapon type. Get one easy win in one of the Extremus fights, then either deliberately lose in the Insulae (do not revive your warrior), or just delete the gladiator. There you go, a slot has been freed up for the next fighting style.

Just to recap the styles:

Sword & Shield
Unarmed (just unequip any fighter for this style)
Dual Daggers
Dual Swords
Two-Handed Sword
Two-Handed Hammer
Two-Handed Trident
Spear & Shield

You will probably find Spear and shield and two handed Trident are the last to get out of the eight you need, I had to get my fame to rank 10 before Spear & Shield unlocked, but if you recruit Oenomaus, you might be able to get it sooner.

ps. I can confirm you don't need to keep all the fighters with their different styles to get the achievement.

Just to add to what's already been said: you can use the very first fight (the "practice" fight against the sword and shield guy) as your one win with any style and it will still count, in case you're having trouble fighting a higher level person. It worked for the achievement and also lets you get more familiar with each style's combos without having to worry too much about losing the fight.