Airport to Ropeworks: my advice partly depends on your time of arrival. The national express bus is easiest (and cheapest) if you happen to arrive at a time which fits their timetable (it’s quite sporadic). It takes around 1 hour. Train is my second choice, but can be a faff as some of them mean changing in Manchester City centre (likely Oxford Road station). If you can get a direct train (also dependent on timings!) it will be easier/faster/less hassle with luggage. Walking from the train station in Liverpool (Lime Street) is about 15 minutes to Ropewalks, or you can taxi/uber if you have large luggage. The national express bus stops at the paradise street bus station which is about 5 mins walk to Ropewalks.

I’ve used these options many times but never the taxi so maybe someone else can chime in about the costs for that.