One problem related to engine cooling system has been reported for the 1998 Audi A4. The most recently reported issues are listed below.
In March 2003, the oil warning light came on. Dealer diagnosed a faulty sensor that was replaced for $300. In June 2003, the oil warning light again came on. This time it was diagnosed as a faulty oil pump that was replaced at $900. In September 2003 the engine overheated. Diagnosis: faulty thermostat and sensor. Replacement cost $750. Also, washer fluid container cracked and hand brake lever handle cracked (at different times) -- both made of cheap qualtiy plastic.