Is upgrading fro snow leopard to mountain lion will slow down the computer?

If should not if your Mac meets the requirements for Mountain Lion.

Yes, but not so much that you would notice. Older models like yours do not have the horsepower needed by Lion and later versions of OS X, so they do not perform as fast as they do with Snow Leopard.

Upgrading to Mountain Lion

To upgrade to Mountain Lion you must have Snow Leopard 10.6.8 or Lion installed. Purchase a redemption code at the Online Apple Store: OS X Mountain Lion. Mountain Lion is $19.99 plus tax. Use the code to redeem a download of Mountain Lion from the App Store. The file is quite large, over 4 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.

OS X Mountain Lion - System Requirements

Macs that can be upgraded to OS X Mountain Lion

1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later.

2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model

Identifier 5,1 or later.

3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later.

4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later.

5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later.

6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later.

7. Xserve (Early 2009) - Model Identifier 3,1 or later.

To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.

Are my applications compatible?

See App Compatibility Table - RoaringApps.

Note: You should consider upgrading to Yosemite (a free upgrade), the current OS X version, in lieu of Mountain Lion (a 2+ year version that costs $20).

Yosemite will make better use of your RAM (the amount of which, you haven't provided) and has many other under the hood advantages over Mountain Lion.

That's not necessarily true! If the "improvements" of an newer OS uses more resources of the computer, the computer will run slower.

The only exception is if the OS does a better job of utilizing the existing computer's hardware. But usually an OS is released to take advantage of current models' hardware, not older models (even if the older models technically meet the minimum hardware requirements).