Learn Russian: Этот бегун старше, чем мой брат.

Yes, the basic adjective is старый. However, Russian has irregular short forms for comparing things. When comparing the ages of people or living things, you must use the irregular form старше (older). There is another comparative form, старее, but it is generally only used for inanimate objects.

Can I write this comparison without using the word чем (than)?

Yes! You might remember that Russian has two common ways to make comparisons. You can use чем followed by the Nominative case (чем мой брат), or you can drop чем entirely and just use the Genitive case for the second person: старше моего брата. Both are perfectly natural and mean exactly the same thing here.

Where is the verb in this sentence?

There isn't one! As a quick refresher, Russian almost always drops the present tense of the verb "to be" (is, am, are). The sentence literally reads "This runner older," which naturally translates to "This runner is older."

Does the noun бегун (runner) come from the verb бегать (to run)?

Exactly. It is built from the same root as the verbs of motion бежать and бегать (to run). The ending -ун is sometimes added to verb roots to describe a person who frequently does that action.