Well, for starters, let's assume you're renting a 1U dedicated server in a datacentre for $99.99/mo. Your server has 16GB RAM, a quad core 2.8GHz intel xeon and a 1TB HDD, with a 1gbps uplink.
You pay approximately $0.01/gb incoming traffic, and about $0.03/gb outgoing traffic.
You'd be paying $0.60/day when not in games peroid, and about $3.15/day when in the games period.
You're going to be paying 90x $0.60 and 20x $3.15 which is a total of $117 in terms of bandwidth. Including incoming traffic may get you to around $125.
Your server would be running some form of Linux (eg CentOS), so you do not pay for the OS. To further save money, you use php5 and nginx for your web server and web language of choice. These are both free. However, since you are not setting it up, expect to pay around $100 for some IT technicians to do it for you. Since you are also looking for a managed solution, you would probably pay another $50/mo.
In terms of security, you don't need too much. You could probably get your IT technicians to change the SSH port, enable RSA keys for SSH, disable root login, and add iptables rules to protect further for another $100 or so.
You can do the calculations yourself. If you are looking for totally in-house solutions, the server would cost around $399 to purchase and maybe a static $59 per month to keep the server up 24/7.
These are all estimates though, so don't take them seriously (do your own research)