TOKYO—Japan's incoming prime minister, Shinzo Abe, on Tuesday appointed Shigeru Ishiba, a former rival for leadership of the Liberal Democratic Party and a party heavyweight, to the party's No. 2 post, a key role in what will be a major challenge for the new administration—winning upper-house elections this summer.

Mr. Ishiba, who recently made headlines with aggressive comments calling for a weaker yen, will wield considerable influence over the LDP as it returns to power with Mr. Abe, who is set to be chosen prime minister...