Who was the first US president to assume the office after the death of his predecessor?

In 1841, William Henry Harrison was in office for barely a month before passing away placing his vice president John Tyler as the new president. Harrison had given a long-winded inauguration speech from which he developed pneumonia and never recovered. So far he has been the shortest-serving president in American history. Tyler's surprise quick rise to the presidency was given the nickname "His Accidency."

https://www.whitehouse.gov/about-the-white-house/presidents/john-tyler/

When John Tyler assumed the presidency after the death of William Henry Harrison, there was considerable debate as to whether he should be recognised as a full president or merely an acting president. The U.S. Constitution at the time (Article II, Section 1) stated that the powers and duties of the presidency would "vest" in the vice president in the event of the president's death, but it was unclear whether this meant that the vice president would actually become president or serve in a temporary capacity.

Tyler insisted that he was the legitimate president, not just an acting one. He immediately took the presidential oath of office and assumed all presidential powers and responsibilities. Critics, including some in Congress, argued that Tyler was only acting as a caretaker until a new election could produce a president. They often derisively referred to it as "His Accidency".

The lack of clarity in the Constitution fuelled the debate, but Tyler's decisive action ultimately resolved it. By asserting the full powers of the presidency, he set a precedent that was later codified in the 25th Amendment (ratified in 1967), which explicitly clarifies the line of succession and ensures that the vice president becomes president in such circumstances.

https://en.wikipedia.org/wiki/John_Tyler#Presidency_(1841%E2%80%931845)