This achievement is the hardest one in the DLC, but isn't really all that hard.
In 1998 mode you're supposed to use the stealth approach and non-lethal weapons.
That means that your arsenal is: three gas bolts, four tranquilizer bolts, and five alarm bolts. And your plamids, of course, that work the same as in other modes (with the exception of Possession) but EVE runs out much more quickly. Health is severely reduced and one or two blows will kill you easily. Ammo is scarce, enemies detect you easily and you can't use the hint button (default N on keyboard).
That all being said, this mode isn't a big challenge. The key here is to follow the basic premisse to a T: Be stealthy. You can pretty much avoid conflict most of the time, and although enemies can discover you more easily, the AI is still kinda dumb and if you hide for a few seconds they will go back to normal mode instead of "alerted". In many stages of the game that are full of enemies (an example would be the start of the "back to Columbia" part), you can pretty much knock one or two enemies and sneak (or rush) through the rest of them.
Other general tips are:
-Abuse the Peeping Tom plasmid. The Mod that enables it to not use EVE to see through walls is a must a well, so get it as soon as possible. Possession is less powerful in this mode, but it (and Old Man Winter) can be great for moments of panic when you're overwhelmed with enemies and you can still get money from the vending machines with it. I'd recommend following a guide to find all mods for these plasmids. Ironsides isn't that useful in this mode.
-Save money for lockpicks, since opening certain areas can save you (for example, during the ambush after you go back to rapture, the locked freezing area is a great spot to hold up and pick each enemy that comes through)
-You do have to mind your use of the crossbow ammo, but if you're quick enough you can salvage your tranquilizer arrow back from the body of the downed target. Using the alarm bolt to divert enemy attention is always a useful strategy. Now the gas bolt is best used in addition with the alarm bolt, and that in crucial in certain areas. More on that on the next tips.
-After you get the Hair sample and open the security doors in that area, the place will be swarmed with enemies. This is a great place to dispatch many enemies with the gas bolt. Just use an alarm one and four or more enemies will come close to it. Use the gas bolt, and few will remain. Opening a path for your escape.
-The Manta Ray lounge is probably the hardest area of the DLC. Because of the fire wielding enemy and if you alert one, many will come after. The fire wielding one can be knocked out right at the start. You can either stun him will a tranq arrow or freeze him with Old Man Winter, quickly dispatch of the other enemy with another arrow and run to stunned enemy and knock him down. There will be many enemies left, but use the same strategy of alarm and gas bolt to get rid of many. The rest is all a matter of being careful and using Peeping Tom.
And that's about it. This mode isn't as hard as it seems. This DLC is rather short and you'll be finishing this in almost no time, with not much frustration.