Are countries required to reveal intelligence when they use an intelligence to arrest a government official of another country? For example, if the U.S. arrest a Russian diplomat for spying, is the U.S. required to show the intelligence that proves the diplomat was a spy, or the U.S. can do whatever it wants? Is there any precedent that's similar that shows us what might happen in such a situation?
No. What a country does within its own borders is its own matter.
Under diplomatic conventions, diplomats have immunity from arrest or prosecution. However, diplomats can be expelled without the host country having to give a reason. So diplomats are not arrested for spying, they are just told to leave, and no reason is given. For example, following the poisoning of Skripal, multiple Russian diplomats were told to leave, and many UK diplomats were told to leave Russia. No evidence needed to be presented the countries just gave the embassy a list of names and the diplomats had to pack their bags.
A country's court system may have evidential rules that would require intelligence to be revealed in open court. Some countries have such rules, some don't, but this is an internal matter.