How long would a letter take to arrive in Geneva from Ingolstadt? (Georgian era)

I'll dare to take a shot at this. First, you have to consider that at this time, postal connections would not have been daily, but two or three times a week. That will probably add one or two days to the travel time of postal relay riders. But relay rider services would only exist for the main routes, others would have coaches for taking travellers along. Those are a bit slower, mostly because of longer layovers.

Now take a look at the actual route the letter would take. In Germany, the letter would be transported by the Reichspost - one of the few centralized services that existed on despite Bavaria, Württemberg and Hohenzollern (the main entities en route) being de facto independent states since the 17th century and the dissolution of the German Empire by Napoleon. Originating from Ingolstadt, the letter would be transported to Schaffhausen in northern Switzerland.

From there, it would be transported by the Bernische Post (also called Fischerpost, as it was licensed to the Fischer family). Despite swiss postal services being organised by canton until mid-19th century, all services along the route Schaffhausen - Geneva were licensed to that organisation.

The swiss part is actually easier to figure out, since the route goes mostly along river valleys: first down the Rhein valley to Koblenz (Aargau), then up the Aare river and along the Lac de Neuchâtel southwest to Lausanne and the Lac de Geneve. That part has a length of approx. 350 km. By horse, and without major topographical obstructions the distance should be coverable in three days. But since it is one of the major routes for mail in Switzerland, maybe there were express services available covering the distance in two days.

In Germany, there are two postal routes that can be taken. Wikimedia has a map that shows the Postkurse (postal courses) existing in 1711. It is quite difficult to decipher, but it seems there are two possibilities:

• The first route goes via Augsburg and across the Schwäbische Alb mountains to Lindau at the northern shores of the Bodensee (Lake Constance). I am not sure, but I suspect the letter would have to be handed off at least once, a direct connection between the two cities is not that likely. But even then, I would suspect the 250 km distance could be covered in two days. From Lindau, the letter would have to cross the lake on a ferry to Stein am Rhein. From there to Schaffhausen it is only 20 km, but all in all the lake passage will certainly take another day
• The second route, which is most probably a main line, goes via Ulm to Schaffhausen. Google Books has the scan of a 19th century study about the Sankt Gallen postal serices that contains a longish report about 17th century court proceedings concerning the rights of swiss postal services to cover the route Schaffhausen - Ulm - Nuremberg (which Ingolstadt would have been on). Judging by the fierceness of the competition, it seems to have been a profitable undertaking.
  Anyway, the actual route has a length of about 250 km, but with a major obstacle: the steep slopes of the Black Forrest mountains. I am quite certain that no rider relay would be able to make it in less than three days.

Adding everything up, I come up with this:

1. Waiting for the non-daily service to take up the letter: 1 day.
2. Ingolstadt - Schaffhausen, on whatever route: 3 days.
3. Handover from Reichspost to Bernische Post: 1 day.
4. Schaffhausen - Geneva: 3 days.

That makes a minimum of eight days, if everything goes smoothly.

Another way to get there is with a mile table from 1629 that lists distances along postal routes, starting from Augsburg (the seat of the Thurn und Taxis family, the Reichspost Postmaster General.). For the distance Augsburg - Geneva, 72 miles are given. (Augsburg - Ingolstadt is not listed, it's probably around 8 miles.) These are Bavarian miles of 10.000 paces or 25,600 feet or approx. 7.4 km, so the total is about 600 km, which matches my numbers above.

These miles can be translated to one mile per hour riding time and ten to twelve miles per day, which also would add up to six to eight days excluding handover/waiting times.