Does anyone know if I will be able to run Tyranny on my PC? Specs: i3 3.4Ghz quad core, 4 gig ram and a Gtx1060 3Gb (In the process of upgrading pc)

You are just short on RAM, but the rest is way above minimum requirements.

Easily. I ran it on an ultrabook running an i7-7500U using the integrated HD 620 grapics; 8gigs of ram though. Played it to completion. Turned off AA and ran the game at 720p for constant smoothness, but 900p was attainable with random drops to <30 FPS in demanding scenes.

Only issue I can forsee is the low amount of RAM you have, but since your card has dedicated vram I don't think you will have too much trouble, if any at all.

TBH you should have added 8gb ram long before you added that 1060, but CRPG's are fantastic games for low end systems, even the new ones such as this, POE, and DOS2.