Can flow rate (or pressure) at the end increase if the diameter of pipe increased and then gradually reduced along the way?
Reducing the pipe diameter progressively would not increase the pressure nor flow rate at the tap. The idea is based on a nozzle at the end of a hose giving rise to a high speed jet of water. The nozzle actually reduces the flow rate (volume/time) through the hose and so increases the pressure just before the nozzle. This gives a higher velocity stream but lower flow rate. I assume what you want is a higher flow rate from your faucet.
Notice: I think I am right on my claim in the above, but frankly I am not sure. See edit below*.
You would get a higher flow rate from the faucet if you remove (unscrew) the aerator on the end of the faucet. This is a flow restrictor. Increasing the head would also increase pressure and flow rate.
*EDIT
The most basic formula for laminar (smooth) flow of water in a circular cross section pipe is Poiseuille's equation. To paraphrase, this states that the volume flow rate for an incompressible fluid is equal to the applied pressure times the cross sectional area of the pipe divided by (the length of the pipe times viscosity times 8). This means that progressively decreasing the diameter of the pipe would decrease flow and not increase it.