RoughRomanMemes

while the Russian was borrowed much later when Latin already undergone a number of sound shifts

The changes in the word aren't from Latin, they're Slavic in origin. They show the borrowing was rather early, and a bit messy:

• Latin ⟨Caesar⟩ /'kaɪ̯.sar/ →
• Greek ⟨Καῖσᾰρ⟩ /'kâi̯.sar/~/'cɛ:.sar/ →
• Gothic ⟨𐌺𐌰𐌹𐍃𐌰𐍂⟩ /'kɛ:sar/ →
• Common Slavic *cěsãřь; this should be around ['tsʲĕsa:rʲĭ] in IPA. Eventually shortened to *cãřь, roughly [tsa:rʲĭ]

Then either Russian inherited *cãřь, or side-borrowed it from Old Church Slavonic. Either way the ending yer got dropped, the long vowel shortened, and you get the modern Russian form, ⟨царь⟩ [tsarʲ].

/aɪ̯/→/ɛ:/ could be from Latin, Greek or Gothic; all three underwent it.

That /k/→/tsʲ/ change is the progressive palatalisation of Common Slavic. Something similar happened in Latin, but after Greek borrowed the word, and Common Slavic interacted way more with Greek than with Latin.

But the biggest change was that completely erratic shortening, from *cěsãřь to *cãřь. Wiktionary mentions this happened with English cyning→cyng→king and mistress→miss; I've also seen this happening with Portuguese ⟨senhor⟩ mister, dialectally rendered as "siô", "sô", "nhô" etc. (Plus its female form ⟨senhora⟩→siá, sá, nhá etc.)