Since I live in Chisinau, Moldova which is around 180-200 kms away from Odessa depending on the type of route you take, I decided to take a bus from Chisinau to Odessa. Since I didn't wanted to spend on accommodation, I decided to take a night bus back to Chisinau on the same evening.
At 8 am I took a bus (Marshrutka) from Chisinau (capital of Moldova) to Odessa which took around 4 hours (Including the time at the border control). The bus cost 200 Moldavian Lei (approx 825 Rs for one way).
There is a only one border control between Moldova and Ukraine. At the border control, the Moldovan border officials collected our passports and after 15 - 20 mins, passports were handed back to us by the Ukrainian Border Officials. (passports were stamped by both Moldavian and Ukrainian immigration)