What is the ruling of the Sharia on the intervention of an elder brother to recommend his younger brother for admission to a military academy, knowing that he did not offer a bribe, but rather vouched for him as a precautionary measure against the recommendations of others?

If corruption is so widespread that passing an interview is impossible without intercession, even for the deserving, then your brother is not at fault for what he did, lest the deserving candidate be unjustly excluded. And if the deserving person succeeds and performs their work skillfully and accomplishes it as it should be, then there is no fault in their work or in the earnings derived from it.