Neo: You ever have that feeling where you're not sure if you're awake or still dreaming?
This line is specifically pointing out that Neo doesn't fit in the Matrix. He's one of the one percent (no pun intended) that don't accept the Matrix as presented.
The Architect: As I was saying, she [The Oracle] merely stumbled upon a solution where nearly 99% of all test subjects accepted the program as long as they were given a choice, even if they were only aware of the choice at a near-unconscious level.
-The Matrix Reloaded
Some realize they are in the Matrix, sub-consciously, while others, like Neo, start to see this at a much more conscious level. The Potential Red-Pills.
As to his jumping from one "life" to the other, there is a few reasons for it.
The simplest reason is that Movies tend not to show the mundane things like Neo leaving the club, the train ride, getting home and falling asleep unless there is a pressing need to. Here, it actually makes more sense not to be cause of:
It's thematic. The director wants the audience to feel that Neo sees his public, law friendly persona as different from his private, unlawful one. The straight cut between scenes provides the stark contrast that Neo feels.
Neo's experiencing Existentialism thoughts. He's clearly not happy about his public life. Existential crisis, doubt about his future. Uncertainty about his goals. Hence his' boss' lecture about his lateness.
Jumping from scene to scene is pretty much how Dreams go in real life. You don't always remember the mundane parts, going to sleep, walking to places, etc. Exhibits signs of False Awakening. This is part of Neo's dilemma. He doesn't know if he's awake or not. If he's dreaming. Signs of Lucid and Pre-Lucid dreaming, as Neo is a potential Red-Pill at this point.
The final reason is a bit more complicated. Given the extended universe of the Matrix, as provided by books and the Animatrix, there is signs that life in the Matrix isn't always linear, or in real time. It can be faster, or slower. The humans can experience multiple life times.
In "Goliath", the short story by Neil Gaiman commissioned by Warner Brothers for the Matrix Augmented Reality website WhatIsTheMatrix.com (now defunct), the protagonist character is a human blue-pill in the Matrix. Due to outside interference, his server is disconnected from the matrix, and the machines reboot it. The protagonist experiences three or four different life times, and jumps around, not really experiencing everything, but at the same time he does. At the end, they plug him back into the Matrix, and use his previous "lives" to provide him a better life before he dies. He experiences years in real time seconds.
Of course, As the movies show though, everything in the Matrix seems to occur in real time, especially when the rebels have hacked in, and are talking from the Matrix phones to the Operator in the Real. So Goliath seems to be a special case, but it makes it possible.