Is inflation a possible explanation for what started the expansion? Alan Guth stated in this article

But let me start the story further back. Inflationary theory itself is a twist on the conventional Big Bang theory. The shortcoming that inflation is intended to overcome is the basic fact that, although the Big Bang theory is called the Big Bang it is in fact not really a theory of a bang at all; it never was. The conventional Big Bang theory, without inflation, was really only a theory of the aftermath of the Bang. It started with all of the matter in the universe already in place, already undergoing rapid expansion, already incredibly hot. There was no explanation of how it got that way. Inflation is an attempt to answer that question, to say what "banged," and what drove the universe into this period of enormous expansion. Inflation does that very wonderfully. It explains not only what caused the universe to expand, but also the origin of essentially all the matter in the universe at the same time.

Is it possible that time started with the universe inflating, and inflation was what caused the universe to start off expanding, then after inflation ended, the universe continued to expand from that kickoff ?
An anology would be : a spaceship that was driven by fuel, then after the fuel ran out, the spaceship continued to move from the inertia.
 A: I have to admit that the paragraph you cite does not look like an awfully great piece of scientific writing. (Unless it is followed by a much more understandable explanation of what Guth actually means.)
Let me try to describe the main issue the inflation theory resolved. Scientists observed the sky and measured the Cosmic microwave background in it, the light from the Big bang era. The Cosmic microwave background tells us about the distribution of temperature and density in the Early universe, and it turned out that it contained "too large" coordinated structures. What I mean by "too large" is the fact that when one used the usual Big Bang theory without inflation, the regions were coordinated over distances that could never come into contact. 
Of course, you could imprint these coordinated structure in the initial conditions of the universe. The regions never came into contact, they were just created so as to be coordinated, Gottfried Wilhelm Leibniz, the Enlightenment philosopher and mathematician, might call this a "pre-established harmony".
However, inflation proposes a mechanism by which these large coordinated structures arise. One allows a primordial soup to interact with itself in a natural way in a pre-inflation era and then "claps", and inflates the universe by 26 orders of magnitude. Consequently, the resulting structures in the Early universe are exactly what we see in the cosmic microwave background, and they also fit well with the cosmic structures we see in the present era.
So this is what Alan Guth means when he says that inflation explains how "matter got that way" - he means that it provides a deeper explanation for the state of matter in the cosmos that we are able to observe directly (we are unable to observe any direct messenger from the inflation era).
This is of course not the whole story, the devil is in the details of how the mechanism of the inflation works, so this is what Guth means when he says "Inflation is an attempt to answer that question, to say what 'banged,' and what drove the universe into this period of enormous expansion." The answer that Guth proposes and which is accepted by the majority (but not an absolutely crushing majority) of cosmologists is a specific physical mechanism called spontaneous symmetry breaking.
I think that Guth is overselling a bit in the last sentence you cite: "[Inflation] explains not only what caused the universe to expand, but also the origin of essentially all the matter in the universe at the same time." Inflation definitely is a major actor in of how matter has formed in the Early universe, but I do not see a way how it could be considered its "explanation of origin".

As for your other questions about the "beginning of time", "universe getting started", and "what started it all", I recommend one of my older answers on the topic. 
A: It is mathematically impossible for an inflating state to arise from a singularity, because the exponential function never reaches zero. If the universe started with inflation, then there was no primordial singularity, no arbitrary high energies, etc. However, if the Big Bang did originate from a singularity, then there must have been a brief period of expansion before the inflation began.
Fro more details see: Inflationary spacetimes are not past-complete
