Hi,
My name is Paul Gregory, I'm a software engineer and architect with 30+ years of experience in the business, much of which has been spent working in the video games industry in various capacities. I have worked at Glu mobile, Ideaworks/Marmalade, and most recently as the technical partner in a 2 person indie startup, where I was entirely responsible for all development of the game "Tumbly Rocks", a casual puzzler available on the iOS and Android stores.
Much of my game development time has been spent with Unity, as such I have a deep knowledge of Unity, both from my time as CTO at Kantar Retail VR, and when developing tools and games at Glu and Tumbly Games.
Your project description is very vague, so it's impossible to give any realistic time or cost estimate without significantly more detail, so I've quoted initially in the middle ground for creating a simple, 2D game on Unity with limited external interactions such as advertising, IAP and account management. You mention a backend service that the game would need to interact with, again, more detail here would help to provide a more accurate estimate. I'd be more than happy to revise my bid if you were able to share more detail about the project.
Sincerely
Paul Gregory