Light business jet aircraft pilot reported the loss of the number 2 air data computer. In IMC; the pilot returned to the departure airport and was later informed by the owner there had been previous mechanical problems.

Narrative

Departed ZZZ Runway 04 with instructions to climb out runway heading to 3000 ft. This was my first time flying this specific Beech Premier; and I was single pilot. Two passengers. Was given no information before the flight that there had been any recent avionics work done on this jet. Was cleared by Tower to depart. 800 ft. OVC. Full power; cross checked both Primary Flight Displays (PFDs) at 70 kts.; V1; rotated and climbed out. About 500 ft. into climb; IAS message on my PFD; I looked at co-pilot PFD and saw airspeed at approximately 50 kts. (my side said 165 kts. and accelerating).Climbed into IMC. Informed Tower that I had lost the copilot Air Data Computer; and Autopilot was inoperative; so needed to return to base. Was prompted by Tower but did NOT [request priority handling] (I should have).Tower switched me to Approach. I climbed up to 3000 ft. and was given vectors to the ILS XXR. I had a couple of +/-200 ft. deviations from 3000 ft. while heads down setting up the new destination and approach on the Proline 21. Switched to Tower frequency. Had a low altitude alert on the ILS from Tower about 4 miles out - again; had to look away briefly and drifted a little low on the ILS. Flew a sloppy hand-flown approach - new airplane (to me); had a horn blaring in my headset (remedy: hit the pitch trim hat on yolk and that silenced it); in the soup; and a little adrenaline. Landed uneventfully; deplaned passengers. The owner was one of them; and he informed me that 'we've had a lot of trouble with the avionics since we bought the plane 5 months ago in Country. We just replaced the Multifunction Flight Display (MFD).' I wish that information had been given to me by the head pilot (who was vacationing and unavailable.)Although not asked by Tower/Approach to call them; I looked up their number and called and they said 'no problem...we knew you had your hands full'.1) Should have [requested priority handling];2) Should have requested a higher altitude clear of the clouds and a block altitude; taken a breath; and then come back in for a hand-flown ILS.3) Even though another friend of mine had been flying this plane the previous week and vouched for it being solid; I should have asked if any maintenance had been recently completed.