Kupiansk on Tuesday morning, Ukrainian officials said.
The town in the Kharkiv region was seized by Moscow soon after Russia's invasion 18 months ago, recaptured by Kyiv last September, and under frequent fire again as Moscow tries to hit back in the northeast.
Some residents remain in the town, but regional authorities have ordered a mandatory evacuation of civilians from near the Kupiansk front because of the difficult situation.
Regional governor Oleh Synehubov said the man killed on Tuesday was a guard at a meat processing plant that was hit in the latest shelling. The prosecutor general's office said a 67-year-old man had also been hurt during the shelling.
Reuters could not verify the situation in the town, or reports from the battlefield.
Kupiansk was home to about 27,000 people before the war and is a rail hub about 100 km (62 miles) east of the regional capital, Kharkiv.