A friend of mine is planning a trip to Japan on short notice. He is not a US Citizen, but a permanent resident. Is there a Visa on Arrival or an easy way to get a visa that applies specifically to US permanent residents?
No, Japan only cares about your country of citizenship, they will have to apply for a visa via the usual process. The only thing a green card gets you is the right to apply for a visa at a Japanese embassy/consulate in the USA, and tourist visas are usually processed in 2-3 days.
Japan offers short-stay visa exemptions for tourism. The list of countries that are eligible for visa exemption can be found at Exemption of Visa (Short-Term Stay). The citizenship of the visitor is the only thing that is relevant for Japanese visa exemption (US permanent residency status has no bearing).