Latin did indeed evolve, but this answer will be a bit more complex than a 'yes' or 'no'. There was not one single Latin spoken by all Romans, with there being a somewhat significant difference between 'classical' (aristocratic) Latin, and 'vulgar' (common) Latin. Classical Latin would not have changed much throughout the history of the empire, and this is clear as later Latin authors will make reference to earlier works, suggesting a backwards intelligibility. That being said, the nature of education in the Roman world likely would have meant that students studying Latin would have learned using older texts written by distinguished individuals. This continues into the Middle Ages, with monastic communities often copying earlier Latin texts for use in studying grammar.

Now for vulgar Latin, the issue at hand becomes a bit less clear. We know that even by the late Republic, vulgar Latin was shifting in its pronunciation. For instance, the letter 'h' was beginning to drop from the language. Of course, as you progress towards late antiquity, Latin begins to develop sounds which are more familiar to anyone who knows Veni Vidi Vici as 'Vay-Nee, Vee-Dee, Vee-Chee', rather than 'Way-Nee, Wee-Dee, Wee-kee' (this includes 'u' becoming a fricative [v], and c becoming [ č ] (like the ch in 'chip'). In this period, we see the shift in pronunciation of some diphthongs as well, notably the 'ae' shortening (or, monophthongizing- which really just means it shortens to a long 'e'). Vulgar Latin, at a higher rate than classical I reckon, was also taking on new vocabulary. This was more of a regional change, as regional languages began to blend into the Latin. I am most aware of this happening in Western Europe, with many Celtic and Germanic words being picked up into the new Latin vernacular spoken in the region. However, surely the aristocratic 'late' Latin was doing this as well.

Of course, if we go too far forward, into sometime in the fifth century or so, while Latin does persist, the common, 'vulgar' Latin begins to shift into a new vernacular referred to as 'Romance'. This took many forms, such as Gallo-Romance, which would eventually become French, or Iberian Romance, which would become Spanish. This process began once there was considerable blending of the vulgar Latin in the region with the newer Germanic languages which were introduced in late Antiquity. In fact, one of the authors I study, Anthimus, gives us a glimpse of a pre-Gallo-Romance Latin. How we can tell that it is beginning to shift towards the Romance is by its diction, mostly. For instance, Anthimus begins using the verb mittere (to send) in place of the verb ponere (to put), similar to its usage in the French mètre (to put). Another example is Anthimus' word choice for various local fish, which typically use forms later associated with Gallo-Romance (likely plucked from local Celtic and Germanic languages).

So- for your main question, yes there would have been intelligibility. More so for the elites who were studying a somewhat standardized Latin; however, I imagine that a speaker of vulgar Latin up to the fifth or sixth century could still understand a speaker in the first century (but there would be some difficulty with some of the diction, I would imagine).

As to your follow-up questions:

When you say modern Latin, this may refer to a couple different things. Up until about the 1950's, academics still wrote some volumes in Latin (although admittedly it was certainly falling out of practice by then). This is something referred to as Neo-Latin. Alternatively, I think you may be referring to the type of Latin you may learn in schools nowadays. This is usually the Latin of some of the prominent first century writers, such as Cicero and Vergil, which is Classical Latin. If it is Classical Latin that you're referring to, then yes there would likely be as much intelligibility as there would have been for an educated Roman during the imperial period (assuming you've reached some level of fluency!).

As for your last follow up question, yes a modern scholar (assuming fluency, again) would have likely been able to converse with a medieval monk to some extent. Monks often learned their Latin from prominent writers, again like Cicero and Vergil (call to mind Vergil's prominent role in Dante's Inferno and Purgatorio, since he was viewed as one of the muses for Dante's work- but that's a whole other topic). They would have used a pronunciation similar to what we would today call 'ecclesiastic' Latin (or church Latin), which draws heavily from the shifts in pronunciation seen in Latin through late antiquity, with developments to consonants like c and u, as well as the shift in the diphthong ae (with a few later changes, such as with the letter 'j' being closer to the modern English [as in the 'j' from 'juice'], some time in the high middle ages).