Obuchi's History as Politician

He was elected a member of the House of Representative for the first time in 1963.

Since then he has been re-elected 12 times.

He experienced a lot of official positions :

Parliamentary Vice Minister of Post and Telecommunications
Parliamentary Vice Minister of Construction
Secretary General of the Prime Minister's Office
Secretary of Government Office for Okinawa Department
Chairman of Secretary Council of the House of Representative
Recently:
Vice Secretary of LDP (1984~)
Chief Cabinet Secretary of Takeshita's Cabinet (1987~)
Chief Secretary of LDP (1991~)
Vice President of LDP (1994~)
Foreign Minister of Hashimoto's Cabinet (1997~)
The 18th President of LDP (July 24th, 1998~)
*LDP is majority party so in Japan the President of LDP becomes Prime Minister conventionally
Finally he got the post of Prime Minister on July 30th, 1998.

The Process:

July 13th, 1998 Prime Minister Hashimoto Ryutaro declared his intention to resign. A Stinging defeat of LDP in election for the upper house of Parliament on July 12th let him made his mind

July 21st, 1998 Obuchi Keizo, 61, Foreign Minister
Kajiyama Seiroku, 71, a former Cabinet Secretary
Koizumi Junichiro, 56, Health and Welfar Minister
run for the election of the President of LDP.

July 24th, 1998 Obuchi Keizo was elected the 18th President of LDP.
Result:Obuchi got 225
Kajiyama got 102
Koizumi got 84

July 30th, 1998 Hashimoto resign his Cabinet.
Obuchi organized his Cabinet and become Prime Minister.

References:

"Gendai Nippon Jinmei Roku,1998(Comtemporary WHO's WHO in Japan,1998 vol.1)"
Compiled by Nichigai Associates,Inc.
Asahi News Paper's Shukusatu-Ban No.925,July 1998