I'm trying to understand if Sanksrit during 500 BCE to 1 CE is the same Sanskrit of 1 CE to 500 CE.
I'm not a Sanskrit SME, but here are some basic thoughts:
At Buddha's times the Northwest of the Indian subcontinent was populated mostly by ethnical Aryans (a proto-Iranian tribe of probable southern-Russian descent), and towards the East and the South they were gradually mixing more and more with the local Dravidian people. Consequently the language was closer to Aryan Prakrit in the Northwest and gradually mixed up with local dialects as you move to the East and the South.
The version of that continuum that got to be known as classic Sanskrit was formalized a few centuries after Buddha by a grammarian by the name of Panini.
Panini lived in a region called Gandhara, a place that was a bustling center of education and philosophy from before Buddha's times until after the times of king Ashoka. That's as far North-West in the ancient India as you can go. So the language Panini has standardized was a developed version of Aryan Prakrit.
In the centuries following Panini, this formalized version, now known as Sanskrit, became the gold standard and was used as lingua-franca by most of the educated people in India as far south as Maharashtra. This is more or less the language that Nagarjuna wrote in.
Buddha, on the other hand, lived before all this, in the north-central-to-eastern part of India, so in his region at his time the standard Sanskrit was virtually unknown. The spoken language was a folksy hybrid between the Aryan Prakrit and the local dialects. Today this ancient language is known by linguists as Magadhi, and a very similar language is what we know as Pali, brought to us by the Tipitaka memorizers.
As you can see from this, the comparison is not entirely meaningful. Nagarjuna wrote in a formalized language with strict grammar and rules, while at Buddha's times such language simply did not exist and what existed was a gradient Prakrit stretching from "more Aryan" in the northwest to "more native" in the east and south. The few people who spoke "more Aryan" version of Prakrit at Buddha's times were Brahmins who used it mostly for ceremonial purposes, and people who grew in pure Aryan families in the northwest.
Most of the info in this answer can be found on Wikipedia.
Gautama Buddha lived around 563 - 483 BCE according to most historians. Nagarjuna lived around 150 - 250 CE.
The separation of Proto-Indo-Iranian to Proto-Iranian and Proto-Indo-Aryan is expected to have taken place around 1800 BCE.
Vedic Sanskrit (also known as Old Indo-Aryan) descended from Proto-Indo-Aryan. The Rig Veda is the oldest known composition in Vedic Sanskrit, and is dated between 1500 - 1000 BCE.
Next, we have the Prakrits, the descendents of Old Indo-Aryan. Prakrits are also known as Middle Indo-Aryan. The Prakrits emerged around 600 BCE. But it might have evolved gradually in the community.
The writings of the Ashokan edicts, the Pali suttas and vinaya, the Gandharan Buddhist texts and the oldest parts of the Jain scriptures, were all composed in early phase Prakrit dialects. The Buddha himself is expected to have spoken some kind of Magadhan Prakrit dialect.
Pali is considered to be a standardized hybrid of various real world early phase Prakrit dialects, that experienced a partial process of Sanskritization. It may not be the exact language that the Buddha spoke, but it is very close.
Apart from Pali, Gandhari, another ancient Buddhist canonical language, is also considered an early phase Prakrit.
Early phase Prakrits descended into middle phase Prakrits (from 200 BCE onwards), which became the ancestor to modern day Sinhala, Marathi, Hindi, Bengali, Bihari etc. Buddhist Hybrid Sanskrit (BHS) and Ardhamagadhi Prakrit (the language of the Jain scriptures) are considered middle phase Prakrits. BHS is something like a mix between Pali and Sanskrit.
Panini, an ancient Indian grammarian and philologist, lived around the same time as the Buddha, and standardized Sanskrit. His work defined Classical Sanskrit. Panini's masterpiece was called Aṣṭādhyāyī, and his work surpassed the work done by ancient Greek and Latin grammarians. Due to this, he is often considered the "father of linguistics". What we call "Sanskrit" today, usually refers to Panini's Classical Sanskrit.
In the time of the Buddha, Vedic Sanskrit was no longer understood by the common people and it was only used for religious and royal ceremonial purposes, in the same way Latin is used by the Roman Catholic Church today. The Buddha therefore forbade his monks from translating the suttas into Sanskrit, and preferred his teachings to be taught in the local dialects (i.e. early phase Prakrit). He wanted his teachings to be understood by the common people.
Much later, due to the emergence of Classical Sanskrit, mostly due to the work of Panini, Classical Sanskrit became the main language of literature and philosophy. At this point, it made sense to make use of Classical Sanskrit to write much newer Buddhist works.
Vedic Sanskrit is analogous to Old Latin. Prakrit is analogous to Vulgar Latin. Classical Sanskrit is analogous to Classical Latin.
Looking at various Buddhist scriptures, we can see that the Pali suttas and vinaya of the Theravada Buddhist school, which descended from the Sthaviravada and Vibhajjavada Buddhist schools, is composed in Pali, an early phase Prakrit.
The Mahayana Agamas, today preserved mainly in Chinese, contains mostly the same contents as the Pali suttas.
The oldest of the Theravadin Pali suttas and the Mahayana Agamas are attributed and dated to the Buddha himself. These are the oldest Buddhist works, originally passed down by oral tradition only, just like the Vedas.
The Gandharan Buddhist texts (184 BCE - 3rd century CE) composed in Gandhari, an early phase Prakrit, is attributed to the Dharmaguptaka Buddhist school, a descendent of Sthaviravada.
Apparently, the Sarvastivada Buddhist school composed its later works in Classical Sanskrit. This school was a "sibling" of Vibhajjavada and a descendent of Sthaviravada.
Early works from the Mahasamghika Buddhist school were composed in Buddhist Hybrid Sanskrit, a middle phase Prakrit.
The earliest writings of the Prajnaparamita sutras of Mahayana Buddhism, namely the Aṣṭasāhasrikā Prajñāpāramitā Sūtra, written around 100 BCE to 50 CE, is probably in Buddhist Hybrid Sanskrit. Most Mahayana sutras are written in different registers of Sanskrit. I believe the newer the writing, the closer it got to Classical Sanskrit.
I'm not sure about Nagarjuna's Mulamadhyamakakarika (MMK), but it was probably written in Classical Sanskrit.
The following diagram (from India in Pixels Facebook page) shows the Indo-Aryan language family tree. "Sanskrit" refers to Classical Sanskrit.