To be honest, I don't know about IDM I've never used it.
The reason I'm saying do it like this, is because I know it will work. Its probably not the BEST solution, but it is a solution at least.
1. Open up Internet Explorer and click Tools>Options
2. Under the Connections tab, click "Always Dial a Connection" and set the Default to your Modem. Also make sure that your username and password are included, and your PC is set to "automatically connect" from the logon screen.
3. Download this Batch File to anywhere on your C drive (your User files should be fine, as long as they aren't on a different partition)
4. Open Task Scheduler (Type Task Scheduler into the start menu, it should be the only option)
5. Click New Task
6. go to the Actions tab, then click New set the time options however you want them then click OK
7. go to the Triggers Tab and set it to Run a Program. Under Program/Script click browse and find the batch file.
8 Click Apply and OK
What will happen now, is at the set time, your computer will run my batch (which just tells it to open Internet Explorer) and your PC will open Internet Explorer and will dial the connection automatically.
The end result is you should have an Internet Explorer window and an active Internet Connection.
Internet Explorer should be able to be closed once it's ran, and your connection will remain active (unless you have set it to disconnect on exit)
Also, because I don't know what version of Windows you have, I have made it for 32 bit. This means on 64 bit the batch will run Internet Explorer 64 bit (which doesn't have flash, so you can't do things like watch Youtube videos) However as I said, you can close Internet Explorer and then run your normal Web browser.
Hope it helps
