WHEN President Bush and Japanese Prime Minister Kiichi Miyazawa meet in early January, they will find at least one thing in common: Both men are skilled diplomats whose current popularity ratings at home are low.For Mr. Bush, who visits Japan Jan. 7-10, winning firm promises from a Japanese leader seen as ineffective by both public and politicians could be difficult, many analysts say. In recent visits to Tokyo by United States officials, the more important meeting was not with the prime minister but with Noboru Takeshita, a former premier and kingpin of the ruling Liberal Democratic Party (LDP). Mr. Takeshita played a pivotal role in replacing previous Prime Minister Toshiki Kaifu with Mr. Miyazawa on Nov. 5 in a intraparty compromise. And, according to Japanese press reports, he has rallied LDP support behind a Bush request for more than $1 billion to help build a supercollider atom smasher in Texas. The US has asked Japan to use the January summit for concessions on trade in autos, auto parts, paper, glass, rice, financial services, and computers. In a recent television interview, Bush accused Japan of "stonewalling" on some trade issues. But even Takeshita remains limited by the LDP's quarrelsome disunity and its inability to win over key opposition parties to pass important bills, such as one to allow Japanese troops to serve under the United Nations flag. And with an election for the opposition-controlled upper house just seven months away, the LDP is shying away from political risks. Foreign Minister Michio Watanabe, for instance, admitted last week that Japan will not be able to compromise on its rice import ban due to domestic political constraints. And negative industry reaction has forced the LDP to pressure the government not to implement a US-requested plan for stiff criminal fines against monopolistic practices. Despite Miyazawa's reputation as a statesman and experienced bureaucrat, he has been faulted for the recent legislative failures. He admits a distaste for the back-room politicking of the LDP, which has held power since 1955. But a main reason for the failure of the troop deployment bill is that socialist opposition members threatened to use the debate to probe Miyazawa's past in the stocks-for-favors Recruit scandal. He resigned as finance minister in 1988 after admitting his secretary took stocks from Recruit. Other top LDP leaders, including Takeshita, were implicated in the scandal, and many observers say the installation of Miyazawa as prime minister is a test of whether scandal-tainted politicians can return to formal power.