How can I update apps that I've installed via .apk file on a phone without Google Play Store?
I would not mind downloading the updated .apk files manually, but is there a way to get notified when an update is available for one of my installed apps?
There are several possibilities, depending on your sources:
1: Also see the alternative-markets tag-wiki and follow its links
2: I won't list all the candidates here, as they might be subject to change. The linked lists however will be kept up-to-date.
Disclaimer: For those not aware, and in case I didn't make it obvious enough: those listings reside on my own website.
Using the APKPure app, it is possible to receive Play Store-like notifications about updates.
I frequently use this app instead of the Play Store. It also allows you to download previous versions.
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