death is a preferable alternative to communism

The only sources for him saying this are two Reddit posts from The_Donald that attribute it to him.

It's actually from the Fallout 3 video game.

Hey OP, I'm going to mark this NSFW just to be on the safe side for folks with image preview turned on. The question is fine, just want to make sure no one has that pop up somewhere they don't want it.

TLDR: No, it's incredibly unlikely that Asanuma was ever going to become prime minister.

To place things in context, 1960 was a very eventful year for Japanese politics. The security treaty between the US and Japan that had been signed when the Allied Occupation of Japan had ended was due to expire that year and Prime Minister Kishi Nobusuke had made his primary political goal negotiating a new treaty that was less one-sided. He was successful, but in doing so he sparked the largest anti-government demonstrations in Japanese history. Kishi managed to get the treaty ratified by the Diet but his popularity had plummeted; he resigned immediately after.

You'd think that this would all be great news for Asanuma's Socialist Party, the largest opposition party, but this didn't actually turn out to be the case.

First, the party had been plagued with factional infighting ever since it was formed in 1955 (as a merger of other parties). This came to a head when the party's right wing split off in January 1960 and formed the Democratic Socialist Party. This cost the party a number of seats (and its party president) and shaved off electoral support that it really needed.

Second, the Liberal Democratic Party (LDP; the party in power) was smart enough to get as far away from the security issue as possible. The LDP had always been divided between pro-business and nationalist conservatives but the anti-government demonstrations scared the party enough to hand the pro-business faction permanent control for the remainder of the Cold War. Kishi would be the last openly nationalist prime minister until his grandson Shinzo Abe, the current prime minister (or perhaps Koizumi, Abe's first predecessor).

The new prime minister, Ikeda Hayato, took office in July and announced his signature "income-doubling plan" in September. This was a bold economic policy which he promised would double Japan's real GDP within 10 years. This was extremely popular and, unfortunately for the Socialists, meant that the focus of the electoral campaign was firmly on economic issues, not security.

The Socialist Party only had 20-25% support in the early 60's, remaining pretty consistently 15-20% behind the LDP. Asanuma winning would have been a major upset and there's no reason to think that it would have happened. When the election was held a month after Asanuma's assassination (at a time when it seems likely that the party would benefit from a sympathy vote), it captured only 145 seats to the LDP's 296 (a decrease of 21 seats from the 1958 election).

Follow up to OP, if Inejiro was leader of a Socialist party, would the question not be would Japan have gone down the road to socialism, rather than communism?