I have never had one of the drink. But I checked online and found that the ingredient includes: L-Carnitine, Glucose, Caffeine, Guarana, Inositol,Glucuronlactone, Maltodextrin, Tuarine, Panax Ginseng. Generally there is atimeline for the drink or food to stay in our digestive system:

Mouth: Seconds
Esophagus: Seconds
Stomach: Up to 3 hours (depending on what it is, protein and fat will take much longer to be digested)
Small Intestine: Minutes
Large Intestine: Hours

Most of our digestion happens in the stomach and the small intestine. Our stomach can absorb some nutrients but the majority of the absorption occurs in the intestine. It takes much longer for protein and fat to be digested. If you take in just water, it will take just minutes for it to be taken up and less than half an hour for it to come out. For the monster drink, looks like it is more of a sugar and caffeine based drink. So it will only take minutes for it to be absorbed (means the same as kicking in your point). But since it has Ginseng and other ingredients, it will take, I guess, 2-3 hours for it to go through the whole digestion system. If you would like some more information on the digestion timeline and digestion organ for different types of energy sources, followed is some more detailed information. Hope that will help you.

Carbohydrates:
The digestible carbohydrates are broken into simpler molecules by enzymes in thesaliva, in juice produced by the pancreas, and in the lining of the small intestine. Starch is digested in two steps:
First, an enzyme in the saliva and pancreatic juice breaks the starch into molecules called maltose; then anenzyme in the lining of the small intestine (maltase) splits the maltose intoglucose molecules that can be absorbed into the blood. Glucose is carried through the bloodstream to the liver, where it is stored or used to provide energy for the work of the body.

Protein:
Proteins must be digested by enzymes before they can be used to build and repair body tissues. An enzyme in the stomach starts the digestion of swallowed protein. Further digestion of the protein is completed in the small intestine.Several enzymes from the pancreatic and the lining of the intestine carry out the breakdown of huge protein molecules into small molecules called amino acids. These small molecules can be absorbed from the hollow of the small intestine into the blood and then be carried to all parts of the body to build the walls and other parts of cells.

Fats:
The bile acids produced by the liver act as natural detergents to dissolve fat in water and allow the enzymes to break the large fat molecules into smaller molecules, some of which are fatty acids and cholesterol. The bile acids combine with the fatty acids and cholesterol and help these molecules to move into the cells of the mucosa. In these cells the small molecules are formed back into large molecules, most of which pass into vessels (called lymphatic) near the intestine. These small vessels carry the reformed fat to the veins of the chest, and the blood carries the fat to storage depots in different parts of the body.