Intent: Convert "day" values to Numbers, to be able to run comparison operators against the row.

gistfile1.js

// Data sample
[
  {
   "end":  "23:00:00" ,
   "day":  "5" ,
   "id":  "003b2c9b-fdae-4112-b29b-6081ae138060" ,
   "start":  "16:00:00" 
  },
  {
   "end":  "17:00:00" ,
   "day":  "6" ,
   "id":  "00466316-585f-4578-95ab-99431ed2bece" ,
   "start":  "12:00:00" 
  },
  {
   "end":  "18:00:00" ,
   "day":  "0" ,
   "id":  "067fc39b-2fe8-42d1-a223-533a73998c6b" ,
   "start":  "12:00:00"
  }
]


// Attempted query
r.table("hours")
  .filter({"id":"003b2c9b-fdae-4112-b29b-6081ae138060"})
  .update(function(doc) { return { day : doc("day") + 1 } })

// Result
{
  "end":  "23:00:00" ,
  "day":  "var_21("day")1" ,
  "id":  "003b2c9b-fdae-4112-b29b-6081ae138060" ,
  "start":  "16:00:00" 
}

You may want to convert all the day fields to number first

r.table("hours").update( {
    day: r.row("day").coerceTo("NUMBER")
})

And then run @mglukhovsky query.

If you want to keep strings for the day field, you can use this query:

r.table("hours")
  .filter({"id":"003b2c9b-fdae-4112-b29b-6081ae138060"})
  .update(function(doc) { 
    return { day : doc("day").coerceTo("NUMBER").add(1).coerceTo("STRING") } 
  })