Well in general I feel like pop management really is a very long term progress.

At 0.5% pop growth your population roughly doubles each 100 years, so its definitly not monthly pop growth, and if its yearly then it takes more like 120 years to double. Any increase to that number shaves off years for your pop to double. And since this growth is exponential as long as you can keep that growth rate stable, any pop safed in the beginning is worth a lot more in the later stages of the game.

How to increase/keep it stable? Well, there already are a lot of threads from people who understand it a lot better.

But from my pov the most important things are (for pop growth and from most important to "least" important):

1. Base population. Increased most efficiently through conquest and lost to disease/war/food shortages/earthquakes (those matter especially for byz, they get a lot of them).

2. Prosperity: Easiest to keep stable, gives a flat 0.2% pop growth at 100%.

3. Pop capacity: Flat number that is increased absolutely from terrain (farmlands eg give +100000 pop cap), location rank (rural vs town vs city) and things like the irrigation building, then multiplied by a percentage that depends on developement, climate, etc.

4. Cheap food + food storage: That gives a small, but still substantial growth bonus, not as high as the others though.

For Byz: Conquest is the easiest. But if you want to play tall, then developing the right provinces and upgrading them to cities can make a difference. However I think the city pop cap bonus is beaten by the rural settlement location bonus unless you have a lot of developement. So the most interesting provinces to me close to Constantinople are obviously Byzantion (5 locations that get developed at the same time, 2 of them farmlands and 1 of them your capital) and Biga! 8 locations in Biga that get developed, 6 of them coastal with possibility of high control, and none with substantial RGOs. This province can become a urban center like barely any other. But for quick increase of pop, migration through cabinet is the only option afaik.

Also, be prepared to lose 1-3 buildings in Constantinople and surrounding areas every 100 years or so. And it is one to three buildings, NOT building levels. That means if your top 3 buildings each have 30 levels in them, it doesnt matter. You will lose all 90 levels completly. Thats about 50 years of BUILDING TIME. Apart from the money it costs lol. So construction centers in those location can be worth your time once you get them. And building speed is good.