IME the thing that really sucks a smart phone battery dry is when it's hunting for coverage. If you're in a solid coverage area I'd just leave it alone, but if you're riding in the boonies and you know there's poor coverage, it might be worth putting your phone in airplane mode. Obviously you need GPS on for strava to work.

As others have mentioned, you will use a little data when you upload.

If you are concerned about data usage, go into your phone settings and set up data alerts. Most carriers will also alert you (VZW will send me an alert at 50%, 90%, 95% etc) and the overage charges aren't what they used to be anyway - I think it's $15 for an extra gigabyte for me so I'm not going to freak out about it.

At some point you will figure out that having a dedicated GPS like a garmin will be much nicer, and then you'll have a nice fresh phone to use when you need it.