Does inhaling a drug into the lungs gets the drug into the bloodstream quicker than injecting the drug

Usually this is based upon different factors. When you inject, you can inject in a variety of ways; Intraosteolar (Through Bone), Intramuscular (Through the muscle), Intracardial (through the heart), Subcantaneus (under the skin). These methods depending on what are generally way faster than through inhalation.

One other thing to recognize is that inhaled medication is also fairly fast, and can actually go through mucus membranes. 

The best way is to swollow and hold in your breath