How do kidney cells excrete their own wastes?
I checked my anatomy notes and online figures about the kidney. There is no mention about dedicated arteries for feeding kidney cells. So it does not work the same way as the heart does, where there is a dedicated coronary artery. The kidneys use the same blood vessels for filtering and for nutrition/waste transport purposes too.




Figure 1 - kidney anatomy - source





Figure 2 - nephron anatomy - source


The urea is created from NH4+ and HCO3- in the liver (mostly) and the kidney because of blood pH regulation purposes. It neutralizes the HCO3- created by the lungs from CO2 and OH-.


  The urea cycle (also known as the Ornithine cycle) is a cycle of
  biochemical reactions occurring in many animals that produces urea
  ((NH2)2CO) from ammonia (NH3). This cycle was the first metabolic
  cycle discovered (Hans Krebs and Kurt Henseleit, 1932), five years
  before the discovery of the TCA cycle. In mammals, the urea cycle
  takes place primarily in the liver, and to a lesser extent in the
  kidney.
  
  In chemical terms, urea synthesis is an irreversible, energy driven
  neutralization of the strong base HCO3- by the weak acid NH4+, and the
  average daily excretion of 30 g of urea is equivalent to the disposal
  of about 1 mol of HCO3- per day. Thus, a major function of hepatic
  urea synthesis is to effect this neutralization, without which the
  body would otherwise be confronted by a major load of alkali.
  
  Urea is excreted by the kidney, and is normally present in plasma and
  body fluids at a concentration of 3.0–6.5 mmol/L.



wikipedia - Ornithine cycle
Textbook of Hepatology - Ammonia, urea production and pH regulation - Dieter Häussinger
1984 - The role of ureagenesis in pH homeostasis


The kidney reabsorbs urea in order to concentrate the urine:




Figure 3 - nephron with material transports - urea resorption at the end of the urine creation process - source



  About 40% of the urea filtered is normally found in the final
  urine, since there is more reabsorption than secretion along the
  nephron.



wikipedia - Renal urea handling:
2007 - Critical Role of Urea in the Urine-Concentrating Mechanism


The kidney secretes the urea to the urine, but it absorbs more by the reabsorption than it secreted.


  The kidney freely filters urea at the glomerulus, and then it both
  reabsorbs and secretes it. Because the tubules reabsorb more urea than
  they secrete, the amount of urea excreted in the urine is less than
  the quantity filtered. In the example shown in Figure 36-1A (i.e.,
  average urine flow), the kidneys excrete ∼40% of the filtered urea.
  The primary sites for urea reabsorption are the proximal tubule and
  the medullary collecting duct, whereas the primary sites for secretion
  are the thin limbs of the loop of Henle.



Medical Physiology by Walter F. Boron and Emile L.


So if we are talking about urea, then it is secreted to the urine. If we are talking about other waste products, then e.g. CO2 is certainly handled by the veins.