Ingushetia borders Chechnya, which has suffered two devastating wars since the 1990s, with opposition forces fighting the Russian military for an independent state.
In a separate attack on Friday, an imam of a mosque in the village of Tarki in Dagestan, which also borders Chechnya, was killed when a male attacker fired shots at a crowd.
"An unknown assailant opened fire with an automatic weapon on people headed to a mosque for prayer. One man was killed and one wounded," a police official said.
Azadi Shikhbabayev, a district administration chief, was also killed on Friday morning, after being shot in the head in the Margaramkent district of Dagestan.
Shikhbabayev's driver was also shot, according to the reports from Russian news agencies, and has been taken to hospital with multiple bullet wounds.
The administrator's predecessor was assassinated in an attack in November 2009.
Ingushetia and Dagestan are part of the North Caucasus region which continues to see attacks on Russian military or government targets by groups wanting an independent state.