They don't really mean "I will," but rather "I will have" and "I will be," respectively. J'aurai is the future of avoir - to have. So J'aurai sûrement un quart d'heure de retard literally means "I will surely have a quarter hour of lateness." Here, retard is a noun in French, so you "have" it. Je serai is the future of être - to be. Je serai sûrement en retard d'un quart d'heure means I will surely be late by a quarter hour. Here, en retard is an adverb, so you are late. The difference is probably easier to understand with different examples: J'aurai du temps libre demain = I will have some free time tomorrow. vs Je serai là à midi = I will be there at noon.
As mentioned by Laura, here we have retard as a noun (le retard, meaning lateness), so it's de retard. I guess it's not du retard because it isn't any particular lateness, but simply lateness in general. Or, maybe, it's just an expression, "de retard," like "de plus" or "de rien" or "de facto" (wait! that's Latin!). Oh, well . . .
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