Hage Geingob, 82, died early on Sunday while receiving medical treatment at a hospital in the capital, Windhoek.
A veteran of the country’s independence struggle, Mr Geingob had been diagnosed with cancer and revealed the details to the public last month.
The nation had lost a “liberation… icon”, Mr Mbumba said.
He has since been sworn-in as Mr Geingob’s replacement and will serve in the role until elections due later this year.
“I am not going to be around for the elections so don’t panic,” he said at a swiftly arranged swearing-in ceremony at state house, just 15 hours after the death of the president.
Paying tribute to his predecessor, he said “our nation remains calm and stable owing to the leadership of President Geingob who was the chief architect of the constitution”.
“I take on this heavy mantle cognisant of the weight of responsibility.”