Besides curing long-term illnesses, does the text provide examples of Udi offering immediate relief for more sudden ailments?

Yes, Chapter 33 provides an account of Udi offering rapid relief. A footnote by R. Hari Sitaram Dixit tells of his friend who came to his office complaining of being unable to sleep for eight consecutive days, even while taking a doctor's medicine. Dixit gave him Maharaj's Udi and told him to take it for three days. The effect was almost immediate. As the text states, the very next day, the friend returned to the office and reported, 'Last night I took the Udi and I slept very well throughout the night.' This demonstrates the power of Udi to provide swift benefits for acute problems.