You might not know this, but:
If the degree of the numerator (x+a) and the deniminator (bx+2) are equal, then the asymptote is equal to the ratio of the leading coefficients of the two.
Ask yourself: What is the asymptote's level? It is y=1. Now ask, what are the leading coefficients of x+a and bx+2? How can you express 1 as a ratio of the two coefficients?
This should answer your need to find b.
Now ask yourself: What can I use the other information ( (0,f(0)) = (0,1/2) ) for? Well, you can find a. Try inserting 0 in your function. What do you get? How can you find a from this new information, and the fact that f(0)=1/2?
You should be done by now