> W téchto se dozwédá sjnjch pilné robé, Bohu čjm povinné, bratřjm, wlasti, sobě.
This is almost modern Czech, with j for í and w for v (which was the standard orthography for hundreds of years and would be familiar to any educated Czech).
Take the most famous poem in Czech, in its original orthography from 1836:
> Byl pozdnj wečer - prwnj mág
> Wečernj mág - byl lásky čas.
This is perfectly readable, like your sign.
One of the comments claims the sign is from 1487 - I would find this extremely unlikely. The mention of vlast (homeland) is suggestive of the era of the National Awakening (19th century), as is the orthography, font, and diacritics. (And the overall theme of education for God and country seems pretty anachronistic for 1487.)
If we go back in time, to say the Kralická Bible (1593), we're still well within the realm of readable text to modern Czechs:
> Na Počátku stwořil Bůh nebe a zemi. Země pak byla nesličná a pustá: a tma byla nad propastj: a Duch Božj wznássel se nad wodami. Y řekl Bůh: Buď swětlo: Y bylo swětlo.
Let's go back all the way to the earliest Slavic texts, about 1300 years ago, which largely predate Czech altogether:
> Iskonï bě slovo. Ï slovo bě u boga. Ï bogъ bě slovo.
A modern Czech would understand most of this, with the probable exception of iskonï ([na] konci, but used here to refer to the beginning). They would probably miss the exact tense being used for bě, but guess that it's a copula (they might see it as a portmanteau of bylo and je, which is close enough). This is probably comparably difficult for a Czech as say Chaucer is for an English speaker (ie not terribly), except it is half a millennium earlier than Chaucer.
Your overall point about language change has merit, but not all languages have changed at the same exact rate. Balto-Slavic languages are relatively conservative as a group (many linguists were very taken with Lithuanian when it was first noticed within Western academia, as it is so conservative that it has considerable utility in the reconstruction of Proto-Indo-European itself).
This is probably more detail than you wanted - I hope it's interesting and helpful. :)