If I upgrade my system will my desktop manager change to Unity?

It will upgrade to Lubuntu 15.04, not Ubuntu 15.04. It will not replace LXDE with Unity (The differences between any flavor of Ubuntu is the desktop environment, so literally Ubuntu is the same as Lubuntu, only with Unity instead of LXDE.) Also, upgrading can have good effects, such as:

• Better security
• More features
• More useful stuff
• etc.

So, you should go upgrade it, and 14.10 is old and end-of-life now.

Here's an image to show it's old and end-of-life:

It will update to the newer version of Lubuntu.

The update process doesn't ever change your 'flavour' of Ubuntu. The upgrade path is very safe, but I'd always recommend making a backup of any important data or photos first.