Reusing old i3 MK1 to build new i3 MK3S+  

RE:

To the best of my knowledge the Mk1 frame is narrower than a mk3 and less tall than a mk3, also the Mk1 Y axis is totally different to a Mk3 Y axis, 
the Y Motor is on the opposite side of the Y Belt, to the mk3. 
and the mk1 uses a different controller board and a different voltage power supply. 

I Modified a Mk2 to a Franken Prusa. (nearly a Mk3)

to gain extra Z height in the mk2 frame, I used different Z bottom and Z Top parts, I Found them on Thingiverse, they were intended to restore Z height on a Mk3with Volcano hotend. It was a long time ago... you will have to hunt for the files. 
the width of the frame, is not adjustable, but the mk3 Heatbed will 'Just' fit, with reduced space to the left and right. 
You could consider re printing the Y motor mount (Mirrored on the Y axis.)   or simply pop the pins out of the motor connector, and insert them in the reverse order to make the motor run in reverse. (I reversed my wires.)   (the Mk1 Y Motor mount was a bit weak, I Printed a re enforced version, from Thingiverse     (I Printed mine, before I ever considered 'Franken Prusa'. you may wish to find the file, and print it flipped on the Y access, so that you don't have to reverse the Y motor wiring... )
the Mk3 series Einsy board, will fit inside the Mk1 controller case... BUT. The MK3S+ Controller board, has much better cable management Alternatively there are some enlarged and reversed controller cases, on thingiverse or printables, which may give you an easier life. 

the MK3S+ extruder and X axis parts are probably the best option... (DO NOT CONSIDER THE MK3 Extruder, the old filament sensor was not very good)
the Mk3s or Mk3s+ y belt adjuster, are better than the mk3 fixed belt holder. 
the Mk1 frame Y Axis was 180mm the Mk3 series Y axis is 210mm, I don't recall having to change out the  10mm all thread (Studding) or the smooth rods... but you need to check this.

If I recall correctly, the Frame corner pieces for the Mk2, have a different shape top surface (compared to the Mk1 corner pieces.)  to allow greater Y travel. so you probably need to print and change those.
the Mk2 LCD Housing, is a little lower than the Mk1 LCD Housing, in order to allow the heatbed to pass over the LCD Housing. 

If you replace the Mk1 PSU with a MK3 PSU, you will have to sort the rear PSU Support, because the mk3 PSU is intended to attach to extruded aluminium frame rails on the mk3, and you have all thread frame rails on the Mk1/mk2 frame. 

IF you choose to use a non prusa 24 volt power supply you will probably have to forget about power panic...  (And you will need to create a shroud for the bottom of the PSU, so don't forget to include a brace for the rear of the PSU. )
My Franken Prusa, does not have Power Panic. 

the Spiral wrap used on the mk1 is nasty, and can lead to problems of insulation wear   and wire migration so Use Braided cable wrapping  sheething. 

Don't forget to get a 24 volt heater for the hot end  (Is your hot end an e3Dv6, or and E3Dv6 Lite?   the e3Dv6 Lite is not suitable for a Mk3S series printer

the e3D v6lite, does not have a separate stainless steel heatbreak, instead the end of the heatsink has an M6 thread, which screws directly into the heater block, and the PTFE Tube goes all of the way through the cold end and up to the back of the nozzle (Which limits the maximum working temperature of the hot end!)

Be aware that the MK52 12 volt heatbed  has a different fastener pattern to the Mk52 24 volt heatbed, and the mk52 12 volt heatbed needs a modification to the mounting holes, on the Mk1 y axis platform...  I Changed My Y axis Platform to a Mk3 platform. 

Hope this helps, 

regards Joan