Hello,
It's not possible to give a quote and time estimation without even knowing the shop url.
I'm used to migrate shop version, doing it since 12 years, but it requires some analysis to see the modules you need, discuss about the theme and features you want, check the specific code you did on your shop and that may need to be done again in the new one, hosting configuration, etc...
It's a very complicated job, and if you want a perfect work, it requires someone experimented to do it. I'm that guy.
Send me a private message if you want to discuss about it.
Best regards,
Yann - Prestaplugins