Should the arm in a push-up be at a 45 degree angle or touching the rib cage?

It can be confusing which way to perform a push-up.  Both rib cage and a 45 angle are proper but emphasize different areas.  If your arms are close to your rib cage then your triceps do most of the work. If your arms are at 45 degrees then your triceps and chest share the load. It depends on your goals.  

If you want to build your triceps and the front of your shoulders then rib cage push-ups are for you. If, however, you want to build chest and triceps (and shoulders) then 45 degree push-ups are a better fit.

Form tips to consider:

1. To keep the pressure off the shoulder joint, make sure to keep your shoulder blades depressed and squeezed together. Imagine you are putting your shoulder blades in your opposite hip pockets.  A good way to do this is to imagine you are screwing your hands into the floor, or wall, etc., towards the pinky side of your hand. Your hands don’t actually move though. This will externally rotate your shoulders and depress your shoulder blades automatically.  Try it. From this point your hands can be in almost any position and it won’t hurt your shoulders as long as you can maintain that depressed shoulder blade position.  If you can’t and your shoulders hurt then you don’t have the stability to do that arm position and you should regress to something easier until your strength is built up.
2. Make sure you’re not leading with your head like a turtle. Look straight down and make a double chin.  Your nose should not be the first thing to touch the floor (if you go down that far). This will prevent you from putting pressure on your neck.
3. Keep your glutes squeezed really tight to prevent anterior pelvic tilt (this will protect your low back).
4. Pulling your stomach in towards your belly button will also help with anterior pelvic tilt.

A good way to get into the right push-up position is to start with your chest on the floor, tighten everything up and then go keeping everything tight.

Push-ups are the best way to build upper body strength without weights and rib cage or 45 degree push-ups are good options depending on your goals.