What you need to do to create a continuous surface representing precipitation is a process called interpolation. ArcMap has a number of tools to do this, based on a variety of statistical and sampling approaches. I'd recommend inverse distance weighting (IDW) as a starting point, because it's one of the simplest to use.
The input for IDW is a single feature layer of points (in this case, weather stations). Each point needs to have one value (i.e., mean annual precipitation), and then the interpolation function estimates what the values should be in between the known points.
Importing station data into ArcMap is straightforward as long as you have a location (latitude/longitude) for each one. The tricky part here is that each station currently has many (see note below) values, and you want one map. You can do an interesting but complicated approach after importing the data, or you can summarize the data using another piece of software (I'd use Excel, but many statistical packages would give the same results) before putting it into ArcMap.
Note: One minor consideration is how much time the data covers. Is it daily precipitation for one year? If so, your resulting map will be "precipitation for 2013", not average annual precipitation. If it is 1961-1990 (or some other interval), then you need to group and summarize the precipitation by year.