I came to know that the cosmic horizon is the part of universe beyond which we cannot see. So if the light from 13.7 billion years ago is what we can observe, does that mean the universe is older than that and we can't see it?
No, the oldest light we can see is as old as the Universe$^\dagger$$^\ddagger$. This is simply because light started traveling at that time, so after one year you'd se light that had been traveling for one light-year, after 1 million years you'd see light that had been traveling for a million light-years, and so on. There is (most likely) more Universe beyond the horizon, but light from those regions just haven't had the time to reach us yet.
You might think that the distance to the cosmic horizon today is 13.7 billion light-years, but because the Universe is expanding, the distance is much larger (in fact 43.6 billion light-years).
$^\dagger$Because the Universe was opaque the first 380,000 years, we can't see all the way, but that's a technical detail.
$^\dagger$In some theories, notably those with eternal inflation, the Universe has existed for indefinite time before the Big Bang. In this answer, I take "the age of the Universe" to mean the time since the Big Bang.
The universe might be eternal to the past, as well as to the future. The simplest theory for that is described in "Steady state eternal inflation", formulated by Anthony Aguirre and Steve Gratton, and available free online: It is compatible with observations of the Cosmic Microwave Background, and is centered on a Cauchy surface from which time extends in opposite spatial directions. A similar but somewhat more elaborate cosmology, by Carroll and Chen, is analyzed, together with the Aguirre-Gratton one, by Vilenkin, in his online paper "Arrows of time and the beginning of the universe".
The Borde-Guth-Vilenkin Theorem is usually construed as preventing the "past eternality" of inflation by requiring a balancing of universal expansion by universal contraction, but the references in the last (2003) formulation of that theorem by its three authors include a footnote exempting the Aguirre & Gratton theory from its strictures. Also, the BGV Theorem does not rule out a start of the universe at arbitrarily long times before the start of inflation (quasi-exponential expansion), as hypothesized in Martin Bojowald's 2010 book titled "Once Before Time: A Whole Story of the Universe", which utilizes Loop Quantum Cosmology. Some of the "bouncing" cosmologies, of which the Wiki "Big Bounce" provides an overview, also imply past eternality, and substitute a repetition of "bounces" for the "Big Bang".
What these cosmologists, as well as the three mentioned in the last of my comments below, are trying to do is resolve that contradiction of infinite density in the zero volume of the singularity at which General Relativity has been considered to break down: Because densities in "negative time" might cancel densities in"positive time", the densities of the objects duplicated in either version of the temporal dimension might cancel each other, thereby replacing that contradiction with zero density in zero volume, which makes sense.
I believe that a widespread misconception leading to the belief that the universe, or its massive and energetic contents, are finite is the fact that their limitation to some finite amount would provide the simplest explanation for the fact that the sky is dark at night: However, because of the fact that expansion is different from speed or velocity, and is (consequently) NOT limited to a rate no higher than the speed of light, such accelerations as were witnessed by the Supernova 1A observations in the late 1990's are entirely permissible in General Relativity. The simplest explanations do not necessarily provide the best explanation of scientific phenomena.
We didn't determine the age of the universe by looking at what the farthest object is we can see.
Instead, we looked at closer objects and determined their distance, speed and rate of acceleration. When you know those, you can work backwards and find a time when all those galaxies would have been in the same place, which is 13.7 billion years ago.
When we look at objects really far away we determine the distance using the so called "redshift". One of the furthest known galaxies "GLASS-z13" sits at a redshift of 13.1. According to our models this corresponds to light sent out 13.46 billion years ago.
Note that the model used to connect redshift to distance is constructed in such away that no matter how redshifted an object is that we observe, it can never be estimated to reside at a larger distance than 13.72 billion years. You can check this out using this calculator.
For instance if we were to find an object with a redshift of 20, this would be interpreted as the objected sending out its light 13.54 billion years ago.
A hypothetical object of redshift z=100 would be interpreted as sending out its light 13.7 billion years ago and an object of redshift z=1000 would be interpreted as having sent out its light 13.72 billion years ago.
Answer: Nobody knows if the universe is more than 13.72 billion years old but no light that reaches us can be interpreted as being older than that because of the model we use to interpret the age of the incoming light.