It took me around 3 weeks to get from Lord to Baron, doing more than 50 fetch, smuggle, find black boxes (trade data, ancient artefacts, etc.) and kill missions for at least 5 factions scattered around Cemiess. To be honest, I was not rank farming constantly those 3 weeks, however the number of missions is not exaggerated since I took minimum one to maximum 10 per day in that time span. For that I am certain, because I was trying to get the Clipper, so I remember that I had put an effort to rank up.
Following the same pattern of playing, I made Viscount from Baron in two months, completing almost double the amount of missions, although this time I completed a lot of cash donation and much less kill and find missions. I estimate that I have completed around 100+ tasks for the same number of factions (5-6).
Until now I have not concentrated my efforts in ranking so I can't give exact numbers and durations, but since I made Viscount I stopped trying to rank up so I will try to record my progress from now on and see when I reach the next rank.