The last Ancestry test I submitted was posted on 18th December, and was marked as received on 12th January. It has just gone to DNA extracted a couple of days ago. The tests are only updated as received when they arrive at the US lab, not when they arrive in Ireland. The test before the one I'm currently waiting on took nearly 12 weeks to get a result, from posting date. The current test is showing estimated completion by 12th February, so it's going much faster so far!
As for attempting to find your ancestor. The best advice I can give is to research your tree as far and wide as possible whilst you are waiting for the results. Investigate all the siblings of your direct ancestors in as many generations as possible and try to bring their lines forward by researching their descendants as far as you can. Many of those people may be of little interest to you in relation to your own direct ancestry, but by discovering as many of them as possible you vastly increase the chance that you may recognise the surname of a DNA match, or that Ancestry can pick up a throughline between you and your matches. It's a lot of hard preparatory work, but it will pay dividends. Otherwise you are just going to be looking at a sea of potentially hundreds or thousands of matches, with no idea how many of them might relate to you, and any one of them could be the connection that you need.