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You're working too hard.

Parsing is easier using a regular expression.

Figuring out the mirrored time is easier if you convert everything into minutes first.

Formatting the output is easier using String.format().

private static final Pattern TIME_FMT = Pattern.compile("(0?[1-9]|1[0-2]):([0-5]\\d)\\s*([ap]m)", Pattern.CASE_INSENSITIVE);
private static final int MINUTES_PER_HALF_DAY = 12 * 60;

public static String parallelUniverseTime(String time) {
    Matcher m = TIME_FMT.matcher(time);
    if (!m.matches()) {
        return null;
    }

    int hour = Integer.parseInt(m.group(1)) % 12;
    int minute = Integer.parseInt(m.group(2));
    String ampm = m.group(3).toLowerCase();

    int minutes = 60 * hour + minute;

    //  ^^^ Normal universe ^^^
    //        ** WARP! **
    // vvv Parallel universe vvv

    minutes = MINUTES_PER_HALF_DAY - minutes;

    hour = minutes / 60;
    if (hour == 0) {
        hour = 12;
    }
    minute = minutes % 60;
    ampm = ("pm".equals(ampm)) ? "am" : "pm";

    return String.format("%02d:%02d %s", hour, minute, ampm);
}