EDIT: Thanks for all the informative and enlightening answers!
So, it is very important during any line of historical query to define one's terms as much as possible. So, here, when dealing with typical soldier, that's a question in and of itself. During the late Republic Thermopylae (as a Spartan battle) is mentioned by e.g Cicero and must have been a topic of conversation amongst the learned. But to what degree that story percolated the lower "classes"...well...
Here's the thing, we reach back and pick out specific dates, but the average person back then would have had a broader sense of things. So when we say Thermopylae we mean the Spartans contra the Persians in 480 but by the late Republic there had already been:
352/3 BC Philip vs Thebes (3rd Sacred War)
That's as many as I recall, there were probably more. If anything, the last one listed may have been more relevant to the average Roman. So "Thermopylae" might not have had the same connotations.
As for the Spartans, again remember...they were not really a power in those days, they had subsequently lost to the Thebans at Leukra...they didn't follow Alexander...they failed at winning their independence from the Macedonians etc.
That's not to say they were completely off the imaginative map. Wealthy Roman era tourists would apparently visit the old temple of Artemis Orthia to see the rites, the emperor Julian would send them a letter (along with Athens, Corinth etc) which was a deliberately archaising gesture. There was often an interest in antiquity, but I doubt the average soldier cared.
Hope that helps. As a follow up I'd recommend “Hellenistic and Roman Sparta” by P. Cartledge and A. Spawforth for a generic overview of the period.
Really interesting question! Informative answer from /u/Ireallydidnotdoit but I'm going to come down in the other side of the fence on this one from around the 2nd century BC. My reasons for that are the Hellenising influence on the officer class, the increasing professionalism of the army and the antiquarianism of the Romans in general.
From the second century BC the Romans were mad for all things Greek. At Cynoscephalae the Romans overcame their trepidation about the Macedonian phalanx after a bit of a drubbing against Pyrrhus in the 3rd century, and Greece was opened up to both Roman exploitation and admiration free of any self esteem issues. The Roman officer class especially developed a real taste for Greek culture and military science, certainly enhancing their technical knowledge. As the senatorial classes were still highly respected at this stage and they were key, inspirational figures in the Army, I don't think it's unlikely that the soldiers might have taken some cue from their officers regarding respect for Greek and Macedonian achievements. After this period the Romans certainly were keen tourists for Greece in general, especially Sparta.
From the end of the Second Punic War and the beginning of campaigns in Greece and Macedon, the Roman Army also began to take on a more professional character, as for the first time it had a core of middle class soldiers who seemed to consider soldiering their job rather than their yearly obligation to the state. Professional soldiers in turn develop a more professional institution with its own professional memory, lionising achievements of the past and setting examples for themselves and future soldiers to follow. Of course the Romans had their fair share of great military victories to celebrate, but with the increasing importance of Greek culture in the Army with the Hellenisation of the officer class, I see it as very likely the Romans would have held up a few Greek and Macedonian battles such as Thermopylae as exemplars, though they were probably more interested in outdoing the Greeks out of Roman pride than emulating them.
Finally, the Romans in general had a strong antiquarian streak, upholding the past as an ideal and trying to meet its exemplars wherever they could. Just because the soldier class didn't produce any writings didn't mean they weren't influenced on this matter: they were still Romans. Once again, with the madness for Greek and Macedonian culture that started to overtake the Romans in the aftermath of Cynoscephalae and later Pydna, this Roman antiquarianism would have given them another reason to respect great victories by the Greeks and Macedonians.
All that said, I wouldn't want to overstate this too much. Much of our modern obsession with Thermopylae is a period of much hindsight- in Rome's day Sparta was a shadow of its former self and it may not have started to develop that really brilliant shine is has in hindsight these days. It was a respected place, the Romans especially liked its reputation for discipline as embodied in the agoge, and Spartan troops featured briefly in Caesars army at Pharsalus and played an inconsequential role later at Phillipi, but in the end the Romans would have been much more impressed with Roman achievements than with Greek or Macedonian ones, with the prominent exception of Alexander the Great.
Tl;dr: So yes, by the time of (Julius) Caesar, I think Roman soldiers would have known the story of Thermopylae and respected the ancestral reputation and achievements of the Spartans, but they would have considered themselves much better and probably saw the most spectacular achievements of Roman soldiers (defeat of Carthage, destruction of German peoples, subjugation of Macedon and all the great Hellenistic kings and the major victories in Gaul, etc) as more impressive than anything the Spartans ever did.
Source: J. E. Lendon "Soldiers and Ghosts" is accessible and instructive on Hellenisation of the Roman officer class and Roman antiquarianism.
Adding to the previous good answers, you may find some interest in this thread:
Long story short: Thermopylae wasn't much of a thing until the Renaissance, which would explain why there's no mention of it in the Aeneid, although Sparta is mentioned four times... interestingly, all the references are to Spartan women! And if Virgil, writing not long after Caesar, doesn't find it worth mentioning, it's hard to imagine your 'typical Roman soldier' would have. And don't forget, it was almost 500 years in the past by then, and the Roman military campaigns hadn't even started when the battle was fought. https://en.wikipedia.org/wiki/Roman_Republic#Early_Republic_.28458.E2.80.93274_BC.29
There is an incident in Appian which may be of some interest to this question:
"It is said that at one of their meetings in the gymnasium Scipio and Hannibal had a conversation on the subject of generalship, in the presence of a number of bystanders, and that Scipio asked Hannibal whom he considered the greatest general, to which the latter replied, "Alexander of Macedonia."
To this Scipio assented since he also yielded the first place to Alexander. Then he asked Hannibal whom he placed next, and he replied, "Pyrrhus of Epirus," because he considered boldness the first qualification of a general; "for it would not be possible," he said, "to find two kings more enterprising than these."
Scipio was rather nettled by this, but nevertheless he asked Hannibal to whom he would give the third place, expecting that at least the third would be assigned to him; but Hannibal replied, "To myself; for when I was a young man I conquered Spain and crossed the Alps with an army, the first after Hercules. I invaded Italy and struck terror into all of you, laid waste 400 of your towns, and often put your city in extreme peril, all this time receiving neither money nor reinforcements from Carthage."
As Scipio saw that he was likely to prolong his self-laudation he said, laughing, "Where would you place yourself, Hannibal, if you had not been defeated by me?" Hannibal, now perceiving his jealousy, replied, "In that case I should have put myself before Alexander." Thus Hannibal continued his self-laudation, but flattered Scipio in a delicate manner by suggesting that he had conquered one who was the superior of Alexander."
http://www.livius.org/sources/content/appian/appian-the-syrian-wars/appian-the-syrian-wars-2/
Notice that, in this discussion of great generals, no Spartans are mentioned...