If a quote contains a capitalization error, should [sic] be used? For example, in the following: “But, alas! this kind heart had but a short time to remain such", the t in this would normally be capitalized. When it is quoted in an essay, should it be formatted, “But, alas! this [sic] kind heart had but a short time to remain such"?
First, note that exclamation marks and question marks do not always end a sentence. While it's not common in contemporary English, writers sometimes use these marks after a word, phrase, or clause in mid-sentence. That's the case in your example sentence. In modern writing, alas! might be set off with dashes or parentheses:
But (alas!) this kind heart had but a short time to remain such.
You'll sometimes see something similar with question marks, where each part of a compound question has its own question mark:
Should I stay? or should I go?
There's no need to highlight these cases with [sic], as they are not errors.
For cases where there there is actually a capitalization error, transcribe the quote as cleanly as possible without obscuring the meaning. If you're quoting something with a distinct style (poetry, archaic English, eye dialect, stream of consciousness) then you probably don't need to make note of it at all, since it will be clear that the unconventional English is a stylistic choice and not a transcription error.
If the text contains only a single capitalization error, it's better to simply correct the single letter with square brackets: “[T]his sentence was missing its initial capital.” A minor correction is easier to read than an error followed by a signpost.
Use [sic] only when the text is too difficult to correct with light editing, perhaps because it has extensive errors or because you're not sure how to correct it. That's unlikely for a simple capitalization error.
There are at least three ways of handling the capitalization problem correctly: add “[sic]” after “this”; quote it as is; and replace the t of this with [T], eg:
“But, alas! [T]his kind heart had but a short time to remain such”
I recommend using the second or third approach. Note, it would be inappropriate to silently convert this to This.
Note, for further information about bracketed items in quotes, see: Bracketed Capital Letter, What is the proper use of [square brackets] in quotes?, or How to add contextualizing text to a quotation?.
For more about [sic] see (eg) How do I properly use [sic] for a phrase? Or do I use it at all?, Can you use "(sic)" in other contexts?, What can I do instead of [sic]?, The usage of “sic” in writing, ’[sic]’ to indicate punctuation errors, or Said In Context: A more appropriate alternative to [sic]?.