Is the banjo bolt the same between the Shimano XTR and Shimano Saint rear brake caliper?
edit: The only XTR calipers that used the appropriate banjo bolt for your Saint brake were part of the m960 series, which was released in the early 2000s and had integrated brake/shift controls (push the brake lever sideways to pull shift cable). these levers used the BH96 hose, which is equivalent stiffness to BH-59 and also had a banjo fitting at the lever. Unless you shift by pushing the brake lever, you need a new banjo bolt.
The only banjo bolts compatible with the BR-M820 are the short banjo bolt (with the o-rings, this is part number  Y8H098010. the bolt by itself is  Y8H001500, and the o-rings are  Y8SY28000) or the BR-M8100 spec bolt (Y2RT98010 and  Y2RT05000, respectively). This is from the Shimano compatibility table on &quot;Compatibility between banjo bolt,&quot; here. Your bike shop may have a Shimano hydraulic brake small parts kit kicking around, with a few of these in a plastic container with a bunch of dividers and other things like rotor fixing bolts, retaining washers, and IS caliper mounting shims. Neither of the compatible banjo bolts are compatible with any of the current XTR calipers; the bolts used on those calipers are all too long. both M-810 and M-820 Saint calipers used the same banjo bolt, and it is not used on any XTR caliper with banjo fitting for which I could find an exploded view on si.shimano.com, except for the early 2000s  BR-M965 (IS mount)/BR-M966 (post mount).
The caliper is compatible with standard length banjos (not banjo bolts), but be advised that for heat dissipation and to put more distance between the hose and the hot part of the caliper, Shimano's gravity oriented brakes like Saint usually come with hoses fitted with a long banjo. You can use any banjo-equipped BH-90 hose, but it needs the short banjo bolt. your existing hose and banjo will fit, but consider a new hose kit with the long banjo depending on your riding/braking style.
If you're buying a new hose kit, here's a screencap of the 2021 Shimano genuine parts catalog with the parts numbers for all hose kits that come with the appropriate banjo bolt, because there are many options and availability may vary. Or you can try and get the banjo bolt separately, as above.

On the last part of your question, just to cover the bases:
-Always use genuine Shimano mineral oil, or for other manufacturers' mineral oil brakes, the brake fluid recommended by the brake manufacturer; third party mineral oil brake fluids can and will damage the seals, because mineral oil fluids are not all chemically the same, regardless of viscosity- the tolerances on seals and bladders are tight and Magura or other fluids can cause the seals to swell.
-Follow the service instructions at si.shimano.com, use a torque wrench, and a dedicated bleed kit for Shimano hydraulic brakes- don't use the same bleed kit for different hydraulic systems, like those that use DOT fluid. si.shimano.com is a great resource for Shimano products, and I encourage anybody working on their own Shimano-equipped bikes to familiarize themselves with this resource.
-When bleeding a used brake, start with a gravity bleed to avoid pushing dirty fluid up through the lever. There is also a procedure to expose and clean the pistons, one piston at a time, using a modified bleed block.