My home is over 50 years old and has cedar shingles on it. Can I paint these shingles?

Would suggest that you have all the shingles pressure washed prior to painting. Your shingles just might turn out to be rather nice and you decide to keep as is. In any event, to properly prep the shingles they need to be squeeky clean or the paint will not stick properly and you'll have to keep repainting quicker than you'd like to. It's worth it in the end for the pressure wash.

After the pressure wash the shingles must dry before you coat them with a good primer and then two coates of topcoat.

I'd stain that cedar after the pressure wash. This link is a excellent product for this.

Sikkens: Simply Beautiful Woodcare Products For Your Wood Home

You asked about paint so we answered about paint. I'd have to agree with amricca that stain would be a better choice. I would go so far as to recommend that you consider Thompson's® WaterSeal® Waterproofer Plus Clear Wood Protector - ThompsonsWaterSeal.com This stuff can come in different tints and can be sprayed on with a garden sprayer, about $68 for 5 gallons.