Yes you can do that.The Pages shelf lets you break a view into a series of pages so you can better analyze how a specific field affects the rest of the data in a view. When you place a dimension on the Pages shelf you are adding a new row for each member in the dimension. When you place a measure on the Pagesshelf, Tableau automatically converts the measure into a discrete measure.
There is a way to do this, but to be honest, you should really be pivoting this data. A format with a single date column generally allows for much easier analyses down the line.
Another option would be to have a Parameter which contained each month (January, February, March, etc.), and then have a calculation which looked something like this (I'm assuming your columns are the name of a month in this example):
CASE [Parameter You Just Created]
WHEN "January" THEN [January]
WHEN "February" THEN [February]
WHEN "March" THEN [March]
WHEN "December" THEN [December]
With this calculation, your first line references the parameter, the values in quotes reflect the text values of the parameter, and the fields to their right reflect the column names of your various months of data. You can then just put this calculation in your view, show the parameter control, and your end user can change the parameter selection to see a different month.
Again, my best advice would be to pivot the data.