I was in a similar situation and decided to look at ready-made companies rather than registering a new one from scratch. The main thing I learned is that the existing history needs to be checked carefully. I wanted to see the tax and reporting history, possible debts or liabilities, previous activities, shareholders, management board and whether the VAT status was still valid.
I also compared several options and eventually used the service described here: ready-made company in Estonia. The term Estonian shelf company (OÜ) came up often during my research.
For me, the important part was understanding exactly what was being transferred and what documents I would receive. I also checked that the ownership and management changes would be properly registered. In Estonia, changes to shareholders and board members have to be reflected in the official register, with the required documents and signatures.
As for timing, I wouldn’t assume that every purchase takes the same amount of time. It depends on the documents, authentication method and registry processing. I planned some extra time rather than relying on an exact same-day transfer.