You can hunt tahr year round, however, there are restrictions around aerial access to some areas.
The winter is the most difficult time of year to hunt tahr with climate and day length working against you. However, if you want to get a bull in the best condition then there is no substitute for the rut in May/June.
In spring time tahr are coming out of winter coat. They are a lot lighter in colour and easier to spot. Also at this time of the year tahr descend to low to mid altitudes, meaning you don't have to climb so long or high.