If you go up the bike near the shed and onto the deck with the black ants, you'll see pot plants and a garden hose on the wood pile side. The Mantis is inside the hose if you follow it up to the plants and slip into the break in the hose.
To summon the Mantis, you need the Kebap. The recipe is in the Stump Lab and requires you to kill the Hedge Broodmother for 5 chunks, along with some splinters and 2 fire ant heads. To access the Stump lab there is a circuitous path well documented involving beating the Assistant Manager and going to a hidden outpost in the Koi Pond (Grounded wiki has explicit instructions).
Now, combat strategy:
to begin, you want healing items, a fair amount. I'd say about 5-10. Now the best part about this is you can just create "questionable slop" smoothies using random ingredients, that gives you a one-off health buff and that's it. Supplement with the Brussel Sprouts or Chewing Gum to increase the effectiveness or give you heals-over-time.
I won first time using the following kit (Solo):
Full Roly Poly armour (level 6-8 in various parts, specced to Bulky)
Antlion Great Sword Level 9 (Mighty)
40 Salt Arrows (requies feather arrows + salt shards)
Black Ox Crossbow Level 5
Mutations:
Meat Shield (Extra HP)
Buff Lungs (Extra stamina)
Blademaster (Sword stamina reduction)
Coup De Grass (Increase crit chance)
Cardio Fan (Increase stamina regen, decrease exhaustion time)
When you summon the Mantis, try to keep him opposite the centre stump and just open up with your salt arrows, they will be highly effective for the first 50% or so of his health bar, about 20 odd of them via the crossbow should drop him close to half.
Once his health drops, he'll start doing lunging attacks at you, at this point you can either try to keep back-pedalling and shooting, or just switch to the greatsword and start slashing.
the mantis has two close attacks, he'll draw back each arm and slash at you once, these can easily be blocked to get a few good extra hits in. If he starts winding up, it'd going to be an AOE type attack, either a Scream that'll stun and drop you for a moment, or it'll be 3 quick heavy slashes in a row that you cannot block. If you see the Mantis draw back, then get ready to dodge.
If the Mantis has hit your health, drink a smoothie to get some health back.
He should drop pretty quickly, if you stay close to him he won't drop the pollen attack so you dont have to worry.
I had mixed levels of health, stamina and regen buffs thanks to milk molars too, so compensate for that by having more smoothies if you dont have many boosted levels.