I decided to buy myself an implant once I won 4000 caps at the Atomic Wrangler, but I don't know which one to pick. Here are my stats:

Strength: 7 (6+Training)
Perception: 6
Endurance: 8
Charisma: 1 (I don't want to increase it)
Intelligence: 5
Agility: 8 (7+Small Frame)
Luck: 7

I play as a (kinda) sneaky Sniper that uses unarmed in close quarter combat (Pushy) .I'm using a hunting rifle with a scope, but I do not wish to use a sniper rifle. I may want to use the ARM later on in the game.

I want to increase my Repair, Lockpicking, Guns, Unarmed (but less than my guns skills) and Speech.

It depends on what you prefer. If you are going to use the AMR later then I'd suggest getting the increase in Strength which will also help with your Unarmed damage.
Perception of 6 is fine for perks and ED-E makes high Perception redundant.
Endurance of 8 isn't really required for a Stealth/Sniper build so I wouldn't raise it anymore.
Charisma of 1 is fine.(I really wish they would have given it more uses)
Intelligence of 5 seems a bit low, my personal opinion is that the points you used in Endurance would have been better suited here.
Agility is good for all of the useful Agility based perks.(Small Frame doesn't actually add the extra point to your base score, its simply a plus one. Perk requirements only take note of your base SPECIAL.)
Luck is the one I personally suggest you raise. The Implant plus the Lucky Shades brings it up to 9 which greatly increases your crit chance. This isn't required for a Sneak character, but it would be useful for those surprise encounters.

Luck is simply my preference, but its really up to you which one you want. Though with your Endurance you could get all of these Implants and one of the more expensive ones. Your Endurance number is the same as the amount of Implants you can get. The Endurance Implant is not a free one, if you get it you will still only be able to get 8 Implants.

The only issue is getting enough caps.

I'd say you should get go for Intelligence if your level is relatively low, say, 12 or less. This is so that you can get more skill points from level ups down the road. Otherwise, go for Luck, it'll allow you to clean out the casinos faster, and you can use the money you earn to get even more implants.

You'll want to increase your Strength to 8 if you want to use the AMR, but that's not prudent right now, unless you really depend on your fists to do the talking.

Strength doesn't influence unarmed damage, Endurance does...

Just checked the wiki, my mistake. Still the only thing that affects Unarmed damage is the Unarmed skill. Endurance does affect how high your starting Unarmed skill is, but it will not affect the damage.

briast it improves charisma ya' know

Well, sense you are being sneaky, getting the boost to speed while sneaking would not hurt. Requirment:
Be sure to upgrade your perception to 7 % agility to 6 for the sniper perk. Saves you quite a bit of time.

I am not using VATS, so I do not want the sniper perk.

Anyways, I chose the Strength implant.