Not a big spec., in terms of RAM, especially for running CWP. Dual core helps a lot though.
In making the leap from shared hosting to VPS, you're going to have to learn how to manage a server. CWP can help but it won't replace required knowledge! It is now your responsibility to ensure that the VPS is secure and kept up to date.
Wordpress resources are highly dependent on which extensions, themes etc. are used.
You'll need to fine tune Apache or nginx (run only one of these in 1GB RAM).
You'll also need to tune mysql/maria.
Forget running amavid/clamd/clamav/spamassissin - without exhaustive tweaking these will be a disaster with CWP.
Upgrading to 2GB RAM will make things easier but doesn't take away the need to follow the above points. Perhaps use a different, Wordpress-centric control panel, if that is all you will host.
(I have a backup/failover VPS with single core 1GB RAM, that happily runs a sizable e-commerce application, when the need arises. In quiescent state is has 99 process running; no antivirus/spamassassin)