That's an Android phone, right? You can tell if the phone is currently using the internet connection because there is a little icon in the top right with two arrows to indicate usage, beneath a letter indicating the network type.
As for turning the net connection on and off, I'm honestly not sure where this "Use packet data" option is. On my phone it's just "Settings -> Wireless & Networks -> Mobile Network" to toggle it on and off. There's also a sub menu with settings to disable mobile internet when roaming, and to toggle "always on mobile data" - which basically prevents anything on your phone from connecting to the internet unless you manually tell it to. So you don't need to worry about Facebook or something using up your bandwidth downloading status updates.
As for batteries, online doesn't actually use up that much extra battery, so don't worry about it too much. Though it depends on the type of network and the signal strength. Obviously it depends on the phone too, so I've no idea how long yours lasts.