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Cups:[1]

The custom is to Kasher cups through Miluiy Veiruiy or Hagala.[2] One may not swerve from this custom. [If, however, one knows for certain that the Chametz was never used with hot Chametz. such as it was never used with a hot Chametz drink, and was never washed with hot water in a Chametz sink, and never had cold Chametz, such as beer, soak in it for 24 hours, and it never soaked in the water of a Chametz sink for 24 hours, then it does not need to be Kashered. Practically, however, it is difficult to ascertain the above as well as we suspect that perhaps a crumb of Chametz once fell inside, and therefore it is to initially be Kashered in all cases.[3]]

Bedieved: If one used the cup for a hot Pesach drink without previously Kashering it, the drink remains Kosher [so long as the cup was not Ben Yomo of Chametz use at the time it was used for hot Pesach food].  

Glass cups: Glass cups are not Kasherable, as explained in the previous Halacha.

A Kiddush cup that contains an upper lip:[4] A Kiddush cup may be Kashered even if it contains an external lip.[5]

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[1] Admur 451:72

[2] The reason: As at times Chametz soaks for 24 hours inside the liquid that is in the cup and it is considered like the Chametz was cooked in it. This is certainly possible by wine or [hot tea] of which it is common to dip biscuits or pastries inside of them and perhaps some crumbs have remained in their liquid for the above amount of time. Likewise, some people drink beer in their cups and the beer remains in the cup for 24 hours. Now, although this is only the minority use of the cup, while its majority use is for cold foods, nevertheless, it is subject to the dispute in Poskim as to whether we follow majority or even minority use, of which we conclude that Lechatchila we suspect for the stringent opinion. [Admur ibid; See Admur 451:26-31] Likewise, as times one washes the cup with hot water together with other Chametz, and at times the cup soaks in the Chametz sink water for 24 hours. [See Admur 451:21]

[3] Admur 451:21 regarding the handle of a pot and the same should apply here as well; Admur 451:46 regarding a grinder

[4] Admur 451:19

[5] The reason: As the lip is external and there is thus no worry that Chametz entered inside. [Admur ibid]