Kainerugaba made the announcement in a now-deleted tweet posted on Wednesday night.
Observers have long believed that Kainerugaba, whose previous Twitter outbursts have sparked uproar, including a diplomatic spat with Kenya, was being groomed for the top job.
Although the 48-year-old general has in the past denied claims he intends to succeed his father - one of Africa's longest-serving leaders - he has enjoyed a rapid rise through Uganda's army ranks.
In his tweet, Kainerugaba wrote: "In the name of Jesus Christ my God, in the name of all the young people of Uganda and the world and in the name of our great revolution, I will stand for the presidency in 2026."
He also appeared to take a dig at his 78-year-old father, writing: "How many agree with me that our time has come? Enough of the old people ruling us. Dominating us. It's time for our generation to shine. Retweet and like."
He had earlier said the government was failing to rein in corruption after an investigation accused cabinet ministers and other officials of stealing iron sheets intended for villagers in the impoverished northeastern region of Karamoja.
Following a row last year over a tweet threatening to invade Kenya, Museveni had sought to rein in his wayward son by telling him to stay off Twitter when it comes to affairs of state.
Museveni had apologized to Kenya in October after Kainerugaba, among other remarks, suggested it would take his troops two weeks to capture Nairobi.
Museveni nevertheless defended his only son as a "very good general," after promoting him to the rank despite stripping him of his role as leader of Uganda's land forces.
To many Ugandans, Kainerugaba's position as heir apparent has been obvious with his dizzying climb through the military, but the government has in the past taken a harsh line against anyone discussing the matter.
In 2013, police shut two independent newspapers and two radio stations for 10 days after they published a leaked confidential memo by a senior general alleging that Museveni was grooming Kainerugaba to succeed him.
Kainerugaba's comments on sensitive foreign policy matters have often caused diplomatic headaches for Uganda.
His tweets in support of Tigrayan rebels in Ethiopia angered Addis Ababa, while his thoughts on Russia's invasion of Ukraine and the 2021 coup in Guinea also raised eyebrows.
Museveni's Son Plans Presidential Run
FILE - Then Lt. Gen. Muhoozi Kainerugaba, son of President Yoweri Museveni, attends a ceremony in Entebbe, Uganda. Taken May 7, 2022.
KAMPALA - The controversial son of Uganda's veteran leader Yoweri Museveni, Muhoozi Kainerugaba, announced plans to run for the presidency in the 2026 elections, saying it was "time for our generation to shine."