Approx. how long would it take to make a 4-5 min anime?

For a short of 5 minutes you'd by looking at about 9000 frames of footage.
Animation is typically, traditionally, broken down into "feet" with 16 frames per foot.
A lone professional animator can reasonably/comfortably do between 10 to 20 feet per week, working full time.
That translates to about 6 months to a just over a year, to produce 5 minutes of animation.

BUT............you are talking "anime" style, action adventure........so you are likely talking more than one character, and then you'd be talking effects layers on top of that--such as shadow and highlight layers.   Every character would need their animation, and then a pass for the shadow animation, and then the highlights.
And then the actual effects animation itself, such as explosions, smoke effects, etc.  And then someone has to inbetween all that........so the output could drop to about 3-5 feet per week--which would take about 2 years to properly complete, for a single animator.
A crew of twenty animators and inbetweeners can knock that time-table down to about , maybe 4 months--again, working full time.
But...........again, BUT.........at a MINIMUM pay-rate of maybe $1000 a week per head ( for experienced pros) that is $16K per for that time period, or $320K JUST for animation.  That does not account for pre-prod or post.......which can tack another $100-$150K on to that.

There's shops that claim they can do animation for much less----usually from......."ahem" foreign shores.......but take a close look at what they actually produce. It's almost always NOT "anime", and looks nothing like it.  
You could do the animation in FLASH or HARMONY, but it's not esspecially faster if you have a fast-paced short, with rapid cuts and a lot of new scenic situation.  All the above aspects still apply.

So "feasible" would be your call.