I'm making a shelf above my kitchen sink with intention of putting glasses there to dry off.
How can I treat plywood surface to make it into drainer board? The sink and the shelf are inside oblique corner, so I can't just buy a dish rack.
I'm looking for something like this: 
Whats' the necessary slope and the width of drain cuts?
How can I treat plywood surface to make it into drainer board?
In terms of finish, the only sure way to waterproof wood is with a good film finish. While epoxy finishes, such as used in marine applications, are absolutely waterproof common interior varnish is actually surprisingly water-resistant. So a good coat of oil-based polyurethane (not thin, no pinholes or missed spots, paying particular attention to cut edges) will provide quite reliable long-term waterproofing.
I would in general dissuade you from trying to make something like this from common plywoods which have glue joints of questionable durability. It would be much preferable to make a piece like this from solid wood.
Wood is surprisingly durable in solid form and even untreated softwoods should give you years of service, although of course you should expect staining. Choose a naturally rot-resistant species such as true cedar, white oak, black walnut and juniper I wouldn't be surprised if a piece like this help up for decades without any surface treatment given they can survive more than ten years outdoors with exposure to the elements.
Obviously if there are any glued joints you must use a waterproof adhesive.
Whats' the necessary slope and the width of drain cuts?
I don't think you need any particular angle, sloped is probably enough :-) But we can use the shedding angle for flat roofs as a guide, go for 1/4" over 12" (6mm over 300mm) as the absolute minimum and anything great than that is obviously beneficial.
How can I treat plywood surface to make it into drainer board?
I would recommend against using plywood for this application. The glues in most plywoods don't stand up to water over time and will begin to degrade, leading to delamination of the plywood layers.
Marine plywood would work in this instance, but that stuff is very expensive and not particularly attractive. For regular plywood, a waterproof or water-resistant epoxy coating would be best.
Otherwise, I'd just go with a regular hardwood (like maple) and coat it with mineral oil or salad bowl finish. You'll have to periodically reapply the finish, but it's not a big deal. Note that these finishes are not waterproof, but they will be suitable for this application.
