tl;dr: Having followed Solskjær's managerial carreer since he started out at Molde, I think he has potential, but getting the Man United job now could be too much too soon.
Long version:
Since Molde is the local team where I grew up, and I still follow them during the Norwegian football season, I have followed Solskjær's work in management quite closely over the years. I think he is better suited for a job with a club trying to qualify for the Champions League and at some point challenge for major trophies, than being involved in a relegation battle. He should absolutely not get a transfer kitty, he is dependent on having a sports director of some sort handling scouting and transfers to avoid him forking out cash on mediocre players whom he thinks he kan polish into solid gold. However, he is good at developing talented players, he is just not that good at spotting how talented a player actually is before he has worked with them for a while.
Norwegian football is still very much influenced by the ideas of Egil Olsen after his successes with the Norwegian national side in the nineties. That means a lot of teams which are fairly good defensive units trying to play counter attacking football. Molde under Solskjær has tried to be more possession oriented and it has been obivious that he has moulded the team after Alex Ferguson's style of football at Manchester United.
At Molde he won the league in his first season (2011) and repeated that feat in 2012. He got off to a slow start, losing to a newly promoted team in the first game, but got on better throughout the season. Still, it was a bit underwhelming as nobody seemed to want to win the league that year and Molde benefitted from Rosenborg going through a difficult transition year after the Swedish national team poached their coach the previous season. However, they won the league again in 2012 in far more convicing style.
2013 started disastrously, after the successes of the previous two season meant that several key players left to pursue opportunities in more prestigious footballing nations. The replacements were not as good as Solskjær thought they were, but a spending spree mid season (in the summer transfer window, Norwegian football ran from March to November in those days (now March - December (brrr))) reinforced the team to finish above mid table and winning the cup final. For some reason Cardiff City thought this made him their goto guy to avoid relegation. The decision to buy a bunch of Norwegian sub par players (all represented by Solskjær's agent friend and business partner Jim Solbakken) was disastrous. Adding Kenwyne Jones to the squad was probably not necessarily wrong, but the other incoming transfers were not fit for a relegation battle.
After being sacked by Cardiff in the autumn of 2014, he took a one year hiatus before rejoining Molde. Molde sacked his successor in the summer of 2015, after he had taken them to their first double in 2014. Apparently, he turned in to an idiot six months later. He then started building a new team, which should have won the league last year, but for every game where they played brilliantly, there was a lacklustre performance and needless points dropped, often against sub par opposition.
As many have said, he has done a very good job of being Not Jose Mourinho at Old Trafford. He has also had some well worked gameplans (first half against Spurs comes to mind), and he has also been found out and seemed not to have the tactical nous to pull a counter move out of his sleeve (second half against Spurs). However, he is learning as he goes, he is a clever guy, and he might be a good fit for the United job. As a player, he was always up for it when he got the chance at a higher level, and I think he could very well grow into the role of manager at Old Trafford. But it all depends on the club setting up the right structure around and above him. Whether they can do that, remains to be seen.
