The simple formula (using high speed steel tool) for spindle rotation is 4 times the cutting speed divided by the diameter of the bar. The cutting speed is in feet per minute and the diameter in inches. For steels the average cutting speed is 80 fpm with high speed tooling and 100 fpm for cast iron; brass and aluminum are way higher. With carbide cutting tools the cutting speed can be much higher. This is, of course, a "rule of thumb" method. In your case you should be fine at 125 rpm with carbide and light cuts.
There are a lot of steels that are cold rolled in 4 inch diameter. One of the most popular steels in industry for larger shafts is AISI (SAE?) 1045. This is a medium carbon steel which is fairly tough and somewhat of a challenge on a small lathe. It is difficult to get a really shiny finish on this material, but with patience... It does not thread well, etc.
Another popular group of shafting steels in the 4" size range are the "stress proof" line of steels. There are various brands of this stuff but it generally comes under ASTM A311.
http://www.niagaralasalle.com/product-stressproof.html
These shafting steels are tough but turnable. For our gang of small lathe and mill spindles, they make a good replacement material, especially if one has access to a tool post grinder or a grinding lathe. One of the sad things about the decline of manufacturing and maintenance in the US is the disappearance of the automotive machine shops. It is still possible but not as easy to take a turned shaft to an automotive machine shop and have it ground to very close tolerance with a beautiful finish on a crankshaft lathe.
Our small machines lack the shaft horsepower of the big shop machines and most of us don't run coolant (I have the option of this on my 14" lathe, but I just don't want the mess). I have an mist style unit that I have not hooked up because I still wonder about what ends up in one's lungs. It is not often that I really see the need to flood the workpiece, just a little cutting fluid..
Good luck, just go slow and take light cuts. When you get close to the finished diameter you may wish to use a file and some emery cloth to get a good finish. Just remember to protect the ways against the grit.