Will being mentioned in a report, of which someone else was arrested and charged with a DUI, produce a hit on my FBI name check for immigration?

The USCIS does not reveal to the public all details about the many-faceted background checks it undertakes in immigration application processes, but you should expect that police report to be known through the application process. From your description, it would appear appropriate for you to make clear in the application process that you were not arrested. The security background check process for you may encounter a delay while the government confirms that the police report reflects neither an arrest nor any other detail compromising your eligibility, but ultimately the police report should not interfere with your eligibility for immigration benefits. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.