That's right, you didn't read wrong, it indeed is possible not only to evolve into WarGreymon on your first Digimon, but also evolve into Omegamon with your second Digimon without new game+!
First of all things, always strive to maintain maximum happiness and minimum discipline, especially important for your first Digimon since Agumon's lifespan isn't too long.
Second, don't make mistakes, that's all.
Third, avoid failing in training and always use intermediate training, getting WarGreymon on your first Digimon isn't easy on new game.
Agumon to Greymon is very easy, just need to be very near max happiness with the following stats:
HP - 1000
MP - none, but less than 3000
OFF - none, but less than 120
DEF - 100
SPD - none, less than 80
BRN -none, less than 80
After the evolution, train some more until you can take on Gabumon, first two Vitium fights and Greymon to unlock the Colosseum.
You should get the first challenge from Taichi as soon as you return to the city, winning it should be doable without any further training. It is important to do this as soon as possible so you'll get the second challenge before it is too late to evolve into WarGreymon.
You can fight more fights at Colosseum to get some items and speed up the appearances of Taichi and Yamato?
After selling the Off. Chip, you should have enough money to buy 99 meat to last you for rest of the training with your first Digimon.
Greymon to MetalGreymon is a bit tougher, but still very manageable, you will need the following stats:
HP - 2500
MP - 2000, less than 3000
OFF - none, less than 300
DEF - 300
SPD - 200
BRN - 200, less than 250
You should get the first challenge from Yamato soon after beating Taichi's first challenge, it is good idea to beat this challenge as soon as possible too so you can get the second challenge from Yamato.
WarGreymon fight should be attempted as soon as you have the necessary stats for WarGreymon evolution and enough offence to win Taichi's second challenge, the timing is very important here, so avoid failing training sessions with all your might.
Evolving MetalGreymon to WarGreymon is tough on your first Digimon and requires great training and timing, the following stats are needed:
HP - 5000
MP - 4000
OFF - none, less than 600
DEF - 600
SPD - 400
BRN - 400
You have won the second challenge against Taichi and WarGreymon, have unlocked WarGreymon and evolved into WarGreymon as soon as the fight is finished, good.
Now you have a strong Digimon to win the second fight against Yamato, clear some of the story so you can get Sirloin, the Training Manual, unlock other training spots and perhaps even unlock advanced training upgrades for gyms.
After doing what you could and WarGreymon dies, remember to save the Digimemory, select the Blue Egg and begin your journey towards MetalGarurumon and Omegamon.
At this point you should have plenty of food, money and the Training Manual, so strict training isn't required anymore, but don't make mistakes!
Dodomon to Wanyamon doesn't require stats, but Gabumon needs the highest stat to be:
HP - highest stat, but less than 700 to give breathing room after evolution
Nothing special, continue training and remember to get your food every day.
Gabumon to Garurumon is easy enough:
HP - none, less than 800
MP - 800, less than 1000
OFF - none, less than 80
DEF - none, less than 80
SPD - 80, less than 100
BRN - none, less than 250
Easy indeed, next up is Garurumon to WereGarurumon:
HP - none, less than 2500
MP - 2000
OFF - 200, less than 250
DEF - 200, less than 250
SPD - 250
BRN - none, less than 250
Still very manageable, but then here comes WereGarurumon to MetalGarurumon:
HP - none, less than 4000
MP - 4000
OFF - 400, less than 500
DEF - 400
SPD - 600
BRN - none
Feels great to be this close to Omegamon, but not done yet!
Now you have both MetalGarurumon and a Digimemory of WarGreymon, select the Digimemory of WarGreymon for MetalGarurumon and witness the birth of one of the mightiest Digimon ever!