Is 1,126,850 yen enough for a 90 day stay?

I am looking to go to shinjuku, I am doing cheap lodging because I intend to "live like a king" for the trip. 4-5 star restaurants, some clothes shopping and the like. Breakfast and lunch are just convenience stores and cafeteria stuff, dinner is fine dining, and nights of drinking in Golden Gai. Also a two pack a day smoker, and I drink plenty of liquor. Since I have always wanted to do this trip I want to have the most fun I can, and am curious if this is enough money to get me through.

That's approximately $110 US per day.

I guess it depends on your definition of "live like a king".

86680 yen per week

1500 x 2 x 7 = 21000 meals x twice daily x days
3000 x 7 = 21000 capsule night x days

Probably the bigger issue is living like a king and fine dining at 4-5 star high end restaurants every night. At say 15,000-30,000 yen per night, you budget isn't high enough. Once a week might work...

Still have transport of around 500-1000 yen per day to factor in.

Could you do 90 days for 1.1M yen, yes, but not as you describe it.
For a 90 day stay you might as well rent an apartment - there are some you can get (legally) for a reasonable price including contract fee/signup. You could do pretty well for say 300,000yen (hugely variable.)
Eating out well - the places I have been recently where enjoying lots of good food and drinks 5000-7000 per person. You might need breakfast/lunch so 1500yen should cover that. Say 7000yen/day *90 days = 630,000 yen.
Transport and entry fees - while there is a lot of cheap stuff, you need to get around and 90 days of doing nothing is a bit dull. I would allow 1000 for each (2000 *90 = 180,000 see, it's starting to add up).
Ok, so the above consumes your whole budget.

And then you want to go shopping . . . . .
And drinking in bars can get pricey . . . .
And that also has you in Tokyo the whole time (no day trips - extra cost)
Fortunately smoking is fairly cheap, but still that's another 90,000 yen

I intend to "live like a king" for the trip. 4-5 star restaurants, some clothes shopping and the like. dinner is fine dining, and nights of drinking in Golden Gai. Also a two pack a day smoker, and I drink plenty of liquor

IMO, 12520‰~ per day would be "a wee bit" insufficient for your itinerary.

If that amount is supposed to cover accommodation, I'd say not at all, so forget about it.
If that amount doesn't include accommodation, you could live on it, but not like a king. Assuming 12,520 yen per day, one meal at a 4-5 star restaurant will eat up more than two-day worth of it. I don't even know where you got the idea that that might possibly be sufficient for an extravagant stay...

So you have around 340000yen per month. That is not a lot. There are plenty of Japanese people living with similar amount of budget, but not luxury lives. For all tourists, everything that you do and every moment you spend time in Japan will naturally make you spend more money than locals living there. So you really need watch how much you are spending and make your budget last 90 days. Obviously you can't go to 5-star restaurant every often or at all. You can't go to bar often either. Also have you considered where you want to go for sightseeing and how much it's going to cost you? Remember EVERYTHING cost you money while you're in a foreign country. Be really careful with your budgeting. Unfortunately, it won't be very luxurious trip with this budget for 90 days.

So you have around 340000yen per month. That is not a lot. There are plenty of Japanese people living with similar amount of budget, but not luxury lives. For all tourists, everything that you do and every moment you spend time in Japan will naturally make you spend more money than locals living there. So you really need watch how much you are spending and make your budget last 90 days. Obviously you can't go to 5-star restaurant every often or at all. You can't go to bar often either. Also have you considered where you want to go for sightseeing and how much it's going to cost you? Remember EVERYTHING cost you money while you're in a foreign country. Be really careful with your budgeting. Unfortunately, it won't be very luxurious trip with this budget for 90 days.

First of All,Welcome to japan.This nation is really polite and Friendly.
I will give you rough idea on Tourist view point.
room cost=2600‰~ x90 Days=234000‰~,Smile Hotel Tokyo Nihonbashi,Japan
Standard Semi-Double Room with Bathroom and Toilet
Food cost=1800‰~ x90 Days=162000‰~,unlimited Food and Drinks, per entry cost
Kakiyasu Sanjaku Sanzunbashi Lumine Shinjuku ten Buffet restaurant:In this place you can eat unlimited amount of Food ,Drinks for 1800‰~ 90minutes.
travel cost=Tokyo subway pass 90 Day=45000‰~
Tokyo subway pass 3 Day=1500‰~,Unlimited ride in tokyo
shopping cost=1000‰~ x90 Days=90000‰~,From 100‰~ shop.
Total cost=531,000‰~
Now ,Travel Beyond Tokyo=10000‰~xnumber of trip=(By JR tokyo wide pass)
Tokyo Disneyland,FujiQ Highland,cinema etc=100000‰~
Hope this will help.

First of All,Welcome to japan.This nation is really polite and Friendly.
I will give you rough idea on Tourist view point.
room cost=2600‰~ x90 Days=234000‰~,Smile Hotel Tokyo Nihonbashi,Japan
Standard Semi-Double Room with Bathroom and Toilet
Food cost=1800‰~ x90 Days=162000‰~,unlimited Food and Drinks, per entry cost
Kakiyasu Sanjaku Sanzunbashi Lumine Shinjuku ten Buffet restaurant:In this place you can eat unlimited amount of Food ,Drinks for 1800‰~ 90minutes.
travel cost=Tokyo subway pass 90 Day=45000‰~
Tokyo subway pass 3 Day=1500‰~,Unlimited ride in tokyo
shopping cost=1000‰~ x90 Days=90000‰~,From 100‰~ shop.
Total cost=531,000‰~
Now ,Travel Beyond Tokyo=10000‰~xnumber of trip=(By JR tokyo wide pass)
Tokyo Disneyland,FujiQ Highland,cinema etc=100000‰~
Hope this will help.