To some extent, yes.
It's quite difficult to define what the alt-right is, since few people use that label for themselves. Per Wikipedia, here are some of the common positions of the alt-right:
The alt-right is a white nationalist, racist movement. Part of its
membership supports anti-immigrationist policies to ensure a continued
white majority in the United States. Others call for the breakup of
the country to form a white separatist ethno-state in North America.
Some alt-rightists seek to make white nationalism socially respectable
in the U.S., while others—known as the "1488" scene—adopt openly white
supremacist and neo-Nazi stances. Some alt-rightists are anti-semitic,
promoting a conspiracy theory that there is a Jewish plot to bring
about white genocide; other alt-rightists view most Jews as members of
the white race. The alt-right is anti-feminist, advocates for a more
patriarchal society, and intersects with the men's rights movement and
other sectors of the online manosphere. Alt-rightists generally
support anti-interventionist and isolationist foreign policies
alongside economic protectionism and thus criticise mainstream U.S.
conservatism. Attitudes to social issues like homosexuality and
abortion vary within the movement. Individuals aligned with many of
the alt-right's ideas but not its white nationalism have been termed
Early adopters of the label alt-right (or "alternative right") include Richard Spencer, a neo-Nazi white nationalist, and Paul Gottfried, a "paleoconservative" who has very strong conservative opinions but doesn't seem to be particularly affiliated with white nationalism, anti-feminism, or other such philosophies. But they don't necessarily define what the alt-right is today. I think the definition given previously is a fair broad strokes characterization of the alt-right, so that's mostly what I'll be comparing it to.
So, how do Ben Shapiro's views compare?
Shapiro is anti-Muslim and anti-Arab. He said that "We're above 800 million Muslims who are radicalized – more than half the Muslims on earth. That's not a minority... the myth of the tiny radical Muslim minority is just that: it's a myth." He also claimed that "Israelis like to build. Arabs like to bomb crap and live in open sewage. This is not a difficult issue." This aspect of his views aligns with the alt-right, particularly with respect to their anti-immigration and white nationalist beliefs.
Shapiro doesn't express overt white nationalist views, but does dismiss racism, having suggested that black people don't face more obstacles than white people in the US, and has asserted that racial wealth disparaties are unrelated to racism. He has also advocated in favor of the Confederate flag. These aspects of his opinions align with the white nationalist beliefs of the alt-right.
Shapiro is generally anti-LBGTQ. He's characterized transgender people as having a "mental illness." He believes that same-sex intercourse is immoral and that gay families are worse than families with a mother and a father. Although the Wikipedia definition notes that opinions on homosexuality "vary" among alt-right believers, since LBGTQ rights are heavily associated with the feminist beliefs that the alt-right generally rejects, much of the alt-right is hostile to LBGT people.
He's also strongly opposed to left-wing groups, criticizing, for instance, the membership of university professors in "leftist parties." The title of one of his books literally is about "destroying" leftists in argument. Although this view is obviously not unique to the alt-right, it nonetheless aligns strongly with them.
He is anti-feminist, saying that "radical feminism" (among other things) has "destroyed the foundation of our own greatness." This aligns, obviously, with the anti-feminism of the alt-right. He's also advocated for traditional gender roles, saying, for instance, that "As a society, we have robbed men of their protective missions," another opinion that aligns strongly with the alt-right.
Shapiro is not aligned with the common anti-Semitic stances of the alt-right, which is perhaps unsurprising considering his Jewish background.
He is also not aligned with the common isolationist and anti-interventionist stances. For instance, in 2005 he wrote an article titled "Why the Iraq War is right for America."
So Ben Shapiro's opinions have significant overlap with the alt-right, particularly with regard to racism, anti-feminism and belief in traditional masculinity, and anti-LBGTQ opinions, as well as a fixation on "leftists." They don't overlap as far as outright white nationalism or isolationism and particularly not with respect to anti-Semitism.