The poem is a short, strange piece of play, and its central joke is the title. An "admiral" would normally command a great fleet, but here the whole navy is one small black dog belonging to someone else ("her"). The speaker seems to be at the end of a year, taking stock, and says they finish it "less one lie." That is a modest, honest kind of accounting: not a grand reform, just one falsehood dropped.
The last four lines are harder to pin down. The seas "striking their flags" means surrendering, as if nature itself has given in, while "cast up all winter / on public talk" suggests that the surrender, or the lies, washed ashore as chatter all season. Judging by the imagery, the speaker may be contrasting inflated public language with the quiet, private world of a person and a dog. The mood is wry, calm and a little tired, with gentle humor underneath.