How can I solve this system? (e = natural log)

The natural-log function is denoted by "ln(x)", where "x" is the argument of the log function. Are you perhaps referring to the natural exponential, approximately equal to 2.71828? If so, then...

x * e ^ -(0.2/y) = 4 (1)

x * e ^ -(1/y) = 1 (2)

My understanding of the system is as follows:

. . . . .
. . . . .
You can solve each of these equations for "x=", and then set the results equal, giving you one equation in one variable. Take the natural log of each side, and solve for the value of y. Back-solve for x.

I haven't done the complete solution, but I've gone far enough to confirm the approximate value you've listed for y.