Is it also possible to use the infinitive instead of a verbal noun in this sentence? (Actually in many sentences they seem interchangeable.)
Yes, it is. The exact phrase "Nie mam czasu robić obiadu" gives 81 results in Google, "Nie mam czasu zrobić obiadu" (not that much difference here, but puts the accent on finishing the action of making dinner): 183 results. But the best is probably "Nie mam czasu żeby zrobić obiad" (to make dinner) - 467 results.
"Nie mam czasu na robienie obiadu" gives 150 results, changing robienie to zrobienie makes it 663 and the winner.
My immediate thought was that "na" is being used here in the same way that French speakers use "en".
In British English we might say "I haven't got time to make lunch" - means the same as "I have no time"