Upgrade and jailbreak 8.1 -> 8.4
Yes, not only is it possible to upgrade and jailbreak, it's very easy.

Updating to iOS 8.4


Install the latest version of iTunes on your computer.
Tap here to download the iOS 8.4 .ipsw file (operating system).
Plug your device into your computer.
In iTunes, select your device.
In the Summary pane use the following on the Check for Update box.


Windows: Shift + Click
Mac: Option + Click

Select the .ipsw file and update.


Jailbreaking iOS 8.4


Download the TaiG jailbreaking software.
Connect your device to your computer using the USB cable.
Temporarily Disable Passcode from Settings > Touch ID &amp; Passcode.
Temporarily turn off Find my iPhone from Settings > iCloud > Find my iPhone.
On Windows, launch the jailbreaking application as an Administrator. To do this, right click on the executable file and select the “Run as Administrator” option.
The software will take some time to detect your device. Once you confirm that it is showing the correct firmware version, deselect the 3K Assistant 2.3.0 (TaiG’s App Store), just below Cydia 1.1.18.
Click on the green Start button to proceed.
After the jailbreak is complete, your device will reboot automatically, and you should see the Cydia icon on the Home screen.