How can I make this antique ginger wine recipe using ingredients available today?
How archaic and fun!
I whipped out some Google-fu and found the following for you:
Tincture of Capsicum
You can actually buy this on Amazon: Cayenne Capsicum Tincture 2 Ounces. It's available other places, but I saw prices as high as 2x this. (9 ml ~ 0.3 oz)
Essence of Ginger
This is from a late 19th century Jamaican cookbook (Classic Jamaican Cooking: Traditional Recipes and Herbal Remedies).

Three ounces of freshly grated ginger
Two ounces of thinly cut lemon-peel
Two pints of brandy or proof spirit (white rum)

Just combine and let sit for 10 days, shaking well.
Essence of Lemon
This can also be purchased. You can likely find some lemon extract in your grocery store, if not here is some on Amazon: Flavorganics Organic Lemon Extract, 2-Ounce Glass Bottles (Pack of 3) - note that it's 3 bottles.
Solution of Burnt Sugar
This isn't just typical caramelized sugar, it's burnt sugar. Sugar that has been heated to the dark caramel stage of 370 F to 400 F (188 C to 204 C). The sugar at this point has lost most of it's sweetness, smells burnt, and tastes kind of bitter. It's commonly called caramel color these days. This is what is gives Cola's their dark color. In the small amount of 25ml and in the presence of 2.5 lbs of sugar (wow!) you won't taste it.
My amazon-fu fails here, but have found it available on this site: http://www.spiceplace.com/mccormick_caramel_color.php
It might be simpler to buy this than attempt making your own. Caramel coloring can be a little hazardous to make, it's easy to go too far and burn your sugar, and dumping any quantity of water into 400 F sugar is a sketchy experience. Here is a how-to if you desire though:
http://www.ehow.com/how_5673239_make-caramel-food-coloring.html
Tartaric Acid
This too can be purchased - Tartaric Acid 2 oz. (56 gr.)
Please note that tartaric acid is not the same as cream of tartar. Cream of tartar is an acid-salt in which the tartaric acid is partially neutralized. This site suggests you can use a 2:1 substitution ratio of cream of tartar to tartaric acid. I'm not sure if this will make a difference in your drink because I've never used cream of tartar in such a way or quantity before.
There is also the possibility that a recipe of this age simply used the term tartaric acid to refer to cream of tartar. Who knows?