Thanks for posting.
If you are talking about the bits of dirt buried in the rough spots on the bottle, then I have found that muriatic acid to be effective. However that is not exactly a household product and I would not recommend going out and buying it just to clean this one bottle.
The embossed design is attractive but the case wear (roughness, scrapes, pings) from repeated use really hurts the value.
See this quart example that sold on eBay for $50 but it is in great shape.
https://www.ebay.com/itm/Vintage-1904-Falstaff-by-Lemp-St-Louis-pre-prohibition-quart-crown-top-bottle-/143672757351