I can understand UPS's concerns and I respect the right of the company to establish appearance standards....but they need to be fair and consistent in regards to how those standards are decided upon and enforced.
There are numerous drivers in my building who were hired with visible ink and it was only in '04 that the company finally decided that visible ink ought to be banned. At that point, new hires were made aware of the fact that visible ink was a no-no but those who had been hired prior to that were grandfathered in. The drivers with ink who were hired afterwards are allowed to cover them up with sleeves, gloves, elastic bands etc.
Whether or not you approve of the OP's lifestyle, tattoo choices or attitude is irrelevant. If he is able and willing to comply with the company's appearance standards and cover up his tats, what right does the company have to deny him an opportunity to drive?