On the Via Claudia Augusta you can ignore the wind: over 20 years of observations it has almost never blown along the route as either a tailwind or a headwind. The season and the passes decide. There is exactly one best month, September; summers in the Alps are wet, and in winter and in November the weather all but closes the route to bike touring. You choose your direction by which side you prefer to take the climbs from, and by how you will get home.
Direction: passes and the ride back instead of wind
The cycle route follows the line of the Roman road (strada romana) from Donauwörth in Bavaria through Tyrol and the Fernpass and Reschenpass (Passo di Resia) passes into Italy, across 3 countries. The average wind along the route never exceeds 0.6 km/h in any month (that is the maximum, and it falls in October–December), while hours with a headwind stronger than 15 km/h make up just 0–1% in any month. Riding north to south, the climbs are about 250 m from Biberwier up to the Fernpass and about 500 m from Pfunds up to the Reschenpass. The official route website viaclaudia.org says there is a bike shuttle at every pass, and the return shuttle from Italy to Germany runs every day. It is against that timetable that you should check your direction and the ride back.
Month, stages and length
A good day (giorno buono) means less than 1 mm of rain from 9.00 to 18.00 and a high of 10 to 30 °C. September delivers 64% of such days, rain on 31% of days and an average high of 18 °C. October is slightly worse at 59%, and daylight is shorter. Summer in the mountains is rainy: from May to August it rains on 40–46% of days. May is the wettest month at 142 mm of precipitation; August is the best summer month, but even it manages only 52% good days against 40% rainy days (≥ 1 mm). In winter and in November good days run from 5 to 28% (5% in January, 6% in December), with highs of 2–7 °C.
The route is split into 6 stages, listed in the table below the article. The northern stages from Donauwörth to Merano, including the 140 km stage 3 Landeck → Merano over the Reschenpass, are best in September. The southern stage Rovereto → Ostiglia along the Adige valley (Val d'Adige) and the Po plain is better in October.
The site calculates 722 km along the OpenStreetMap track line from Donauwörth to Ostiglia. The official site quotes about 650 km for the Padana branch to Ostiglia and about 700 km for the Altinate branch to Altino near Venice; the branches split at Trento. The site's track is longer than the official branch to Ostiglia because south of Trento it contains parallel variants and the site counts them all; in the stage table this shows up as the “Rovereto → Rovereto” loop. Wind is taken at a height of 10 m in a grid cell of about 25 km, so the local valley winds of the Alps do not show up in it. All of this is climate for 2006–2025, not a forecast (more detail in the climate block below); the method is described in sources.
Bottom line: ride in September, and you can push the southern part into October; choose your direction by the passes and the shuttle, the wind has no say in it.