How do you figure out how much 1 cup is worth without a measuring cup?

If you are measuring a liquid, any soda or water bottle will have the fluid measure written on its label. Just keep in mind that 8.0 fluid ounces equals one cup. Many soda bottles are 16-ounces which is 2.0 cups.

If you are measuring a granulated solid (sugar), you can calibrate a cup. Again, try to locate an 8-ounce soda bottle. Fill the bottle with water, then pour the liquid into a container/mug. The water will fill to the volume of 1.0 cups. (Of course, you can fill the bottle with the solid, but that is probably not very convenient with solids).