install Moodle 2.2.6 and then import the 1.9.5 classes directly into it? No, that way leads disaster. Teh database itself is very different, the structure of security is different, and so on.
Best bet is to ask your host if you can install your current Moodle then do an upgrade. It might be more tedious and seem a waste of time, but it is not. The change to v2 then to v2.2.6 again might seem a waste of time, but reality is that there is a substantial step up from v1.9.x to v2. This is a defensive strategy, you can test everything works well before taking the next step. From v2 to v2.2.6 is not as large a step up, so if it work in v2, then it should work in v2.2.6.
Of course, there is an alternative that you may think a safer or easier one again. Set up on your local machine an exact duplicate of your current Moodle. Upgrade to the very latest v1.9.19+, that should work with a minimum of fuss. Then do your upgrades from there, when complete, and satisfied everything works, then install your new Moodle to your server, and upload your data via a csv restoration of the database and zipped moodledata folder. The advantage here is you do not upset your production sites and you can maintain a continuity of service. You can also check things every step of the way where you have complete control of every variable in the environment. In the end also, you have an exact duplicate of your Moodle that you can use as a test Moodle for making any kind of change you want before committing it to your production site.